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\begin{document}

\title{ \normalfont {\normalsize \textsc{University of Minnesota, School of
Mathematics} }\\[25pt] \rule{\linewidth}{0.5pt} \\[0.4cm] {\huge Math 4281: Introduction to Modern Algebra, }\\Spring 2019: Midterm 2\\\rule{\linewidth}{2pt} \\[0.5cm] }
\author{Darij Grinberg}
\maketitle

%----------------------------------------------------------------------------------------
%	EXERCISE 1
%----------------------------------------------------------------------------------------
\rule{0pt}{0.3pt} \\[0.4cm]

\section{Exercise 1: Not-quite-all-rationals}

\subsection{Problem}

Fix an integer $m$. An \textit{$m$-integer} shall mean a rational number $r$
such that there exists a $k \in\mathbb{N}$ satisfying $m^{k} r \in\mathbb{Z}$.

For example\footnote{You don't need to prove these.}:

\begin{itemize}
\item Each integer $r$ is an $m$-integer (since $m^{k} r \in\mathbb{Z}$ for $k
= 0$).

\item The rational number $\dfrac{5}{12}$ is a $6$-integer (since $6^{k}
\cdot\dfrac{5}{12} \in\mathbb{Z}$ for $k = 2$), but neither a $2$-integer nor
a $3$-integer (since multiplying it by a power of $2$ will not ``get rid of''
the prime factor $3$ in the denominator, and vice versa\footnote{You would
have to be more rigorous than this in your solution, if you were to make an
argument like this.}).

\item The $1$-integers are the integers (since $1^{k} r = r$ for all $r$).

\item Every rational number $r$ is a $0$-integer (since $0^{k} r \in
\mathbb{Z}$ for $k = 1$).
\end{itemize}

Let $R_{m}$ denote the set of all $m$-integers. Prove the following:

\begin{enumerate}
\item[\textbf{(a)}] The set $R_{m}$ (endowed with the usual addition, the
usual multiplication, the usual integer $0$ as zero, and the usual integer $1$
as unity) is a commutative ring. \newline(You don't need to prove axioms like
commutativity of multiplication, since these follow from the corresponding
facts about rational numbers, which are well-known. You only need to check
that $R_{m}$ is closed under addition and multiplication\footnote{This means
that every $a, b \in R_{m}$ satisfy $a + b \in R_{m}$ and $a b \in R_{m}$.},
and contains additive inverses of all its elements.)

\item[\textbf{(b)}] Let $x\in\mathbb{Q}$ be nonzero. Then, $x\in R_{m}$ if and
only if every prime $p$ satisfying $w_{p}\left(  x\right)  <0$ satisfies
$p\mid m$. Here, we are using the notation $w_{p}\left(  r\right)  $ defined
in Exercise 3.4.1 of the
\href{http://www.cip.ifi.lmu.de/~grinberg/t/19s/notes.pdf}{class notes}.
\end{enumerate}

\subsection{Remark}

Roughly speaking, an $m$-integer is a rational number that can be turned into
an integer by multiplying it with $m$ several times. So a rational number,
written as a reduced fraction, is an $m$-integer if and only if a sufficiently
large power of $m$ can cancel all the primes in its denominator.

The ring $R_{m}$ is an example of a ring \textquotedblleft between
$\mathbb{Z}$ and $\mathbb{Q}$\textquotedblright. It is commonly denoted by
$\mathbb{Z}\left[  \dfrac{1}{m}\right]  $ and pronounced \textquotedblleft%
$\mathbb{Z}$ adjoined $1$ over $m$\textquotedblright.

Note that $R_{1}=\mathbb{Z}$ and $R_{0}=\mathbb{Q}$, whereas $R_{2}%
=R_{4}=R_{8}=\cdots$ is the ring of all rational numbers that can be written
in the form $a/2^{k}$ with $a\in\mathbb{Z}$ and $k\in\mathbb{N}$.

The ring $R_{10}$ is the ring of all
\textit{\href{https://en.wikipedia.org/wiki/Decimal}{decimal fractions}} --
i.e., of all rational numbers that can be written in decimal notation with
only finitely many digits after the comma.

\subsection{Solution}

For each rational number $r$, we have the following chain of logical
equivalences:%
\begin{align}
\left(  r\in R_{m}\right)  \  &  \Longleftrightarrow\ \left(  r\text{ is an
}m\text{-integer}\right)  \ \ \ \ \ \ \ \ \ \ \left(  \text{since }R_{m}\text{
is the set of all }m\text{-integers}\right) \nonumber\\
&  \Longleftrightarrow\ \left(  \text{there exists a }k\in\mathbb{N}\text{
satisfying }m^{k}r\in\mathbb{Z}\right)  \label{sol.ring.m-int.equiv}%
\end{align}
(by the definition of an $m$-integer).

\bigskip

\textbf{(a)} We begin by proving the following claims:

\begin{statement}
\textit{Claim 1:} We have $r\in R_{m}$ for each integer $r$.
\end{statement}

\begin{statement}
\textit{Claim 2:} If $a\in R_{m}$ and $b\in R_{m}$, then $a+b\in R_{m}$ and
$a\cdot b\in R_{m}$.
\end{statement}

\begin{statement}
\textit{Claim 3:} If $a\in R_{m}$, then $-a\in R_{m}$.
\end{statement}

[\textit{Proof of Claim 1:} Let $r$ be an integer. Then, $\underbrace{m^{0}%
}_{=1}r=r\in\mathbb{Z}$. Hence, there exists a $k\in\mathbb{N}$ satisfying
$m^{k}r\in\mathbb{Z}$ (namely, $k=0$). But (\ref{sol.ring.m-int.equiv}) yields
the equivalence
\[
\left(  r\in R_{m}\right)  \ \Longleftrightarrow\ \left(  \text{there exists a
}k\in\mathbb{N}\text{ satisfying }m^{k}r\in\mathbb{Z}\right)  .
\]
Hence, $r\in R_{m}$ (since there exists a $k\in\mathbb{N}$ satisfying
$m^{k}r\in\mathbb{Z}$). This proves Claim 1.]

[\textit{Proof of Claim 2:} Let $a\in R_{m}$ and $b\in R_{m}$. We must show
that $a+b\in R_{m}$ and $a\cdot b\in R_{m}$.

But (\ref{sol.ring.m-int.equiv}) (applied to $r=a$) yields the equivalence
\[
\left(  a\in R_{m}\right)  \ \Longleftrightarrow\ \left(  \text{there exists a
}k\in\mathbb{N}\text{ satisfying }m^{k}a\in\mathbb{Z}\right)  .
\]
Hence, there exists a $k\in\mathbb{N}$ satisfying $m^{k}a\in\mathbb{Z}$ (since
$a\in R_{m}$). Consider this $k$, and denote it by $x$. Thus, $x\in\mathbb{N}$
and $m^{x}a\in\mathbb{Z}$.

Furthermore, (\ref{sol.ring.m-int.equiv}) (applied to $r=b$) yields the
equivalence
\[
\left(  b\in R_{m}\right)  \ \Longleftrightarrow\ \left(  \text{there exists a
}k\in\mathbb{N}\text{ satisfying }m^{k}b\in\mathbb{Z}\right)  .
\]
Hence, there exists a $k\in\mathbb{N}$ satisfying $m^{k}b\in\mathbb{Z}$ (since
$b\in R_{m}$). Consider this $k$, and denote it by $y$. Thus, $y\in\mathbb{N}$
and $m^{y}b\in\mathbb{Z}$.

Note that $m^{x}$ is an integer (since $m$ is an integer and $x\in\mathbb{N}%
$). In other words, $m^{x}\in\mathbb{Z}$. Similarly, $m^{y}\in\mathbb{Z}$.

Now,
\[
\underbrace{m^{x+y}}_{=m^{x}m^{y}}\left(  a+b\right)  =m^{x}m^{y}\left(
a+b\right)  =m^{x}m^{y}a+m^{x}m^{y}b=\underbrace{m^{y}}_{\in\mathbb{Z}%
}\underbrace{m^{x}a}_{\in\mathbb{Z}}+\underbrace{m^{x}}_{\in\mathbb{Z}%
}\underbrace{m^{y}b}_{\in\mathbb{Z}}\in\mathbb{Z}.
\]
Thus, there exists a $k\in\mathbb{N}$ satisfying $m^{k}\left(  a+b\right)
\in\mathbb{Z}$ (namely, $k=x+y$). (Note that with a little bit more work, we
could have also shown this for $k=\max\left\{  x,y\right\}  $ instead of
$k=x+y$; we were just being lazy.)

But (\ref{sol.ring.m-int.equiv}) (applied to $r=a+b$) yields the equivalence
\[
\left(  a+b\in R_{m}\right)  \ \Longleftrightarrow\ \left(  \text{there exists
a }k\in\mathbb{N}\text{ satisfying }m^{k}\left(  a+b\right)  \in
\mathbb{Z}\right)  .
\]
Hence, $a+b\in R_{m}$ (since there exists a $k\in\mathbb{N}$ satisfying
$m^{k}\left(  a+b\right)  \in\mathbb{Z}$).

Furthermore,%
\[
\underbrace{m^{x+y}}_{=m^{x}m^{y}}\left(  a\cdot b\right)  =m^{x}m^{y}\left(
a\cdot b\right)  =\underbrace{m^{x}a}_{\in\mathbb{Z}}\underbrace{m^{y}b}%
_{\in\mathbb{Z}}\in\mathbb{Z}.
\]
Thus, there exists a $k\in\mathbb{N}$ satisfying $m^{k}\left(  a\cdot
b\right)  \in\mathbb{Z}$ (namely, $k=x+y$).

But (\ref{sol.ring.m-int.equiv}) (applied to $r=a\cdot b$) yields the
equivalence
\[
\left(  a\cdot b\in R_{m}\right)  \ \Longleftrightarrow\ \left(  \text{there
exists a }k\in\mathbb{N}\text{ satisfying }m^{k}\left(  a\cdot b\right)
\in\mathbb{Z}\right)  .
\]
Hence, $a\cdot b\in R_{m}$ (since there exists a $k\in\mathbb{N}$ satisfying
$m^{k}\left(  a\cdot b\right)  \in\mathbb{Z}$).

We have now proven that $a+b\in R_{m}$ and $a\cdot b\in R_{m}$. This proves
Claim 2.]

[\textit{Proof of Claim 3:} Let $a\in R_{m}$. We shall show that $-a\in R_{m}$.

We could do this similarly to our proof of Claim 2, but let us take a shortcut
instead: We have $-1\in R_{m}$ (by Claim 1, applied to $r=-1$). Hence, Claim 2
(applied to $b=-1$) yields $a+\left(  -1\right)  \in R_{m}$ and $a\cdot\left(
-1\right)  \in R_{m}$. Now, $-a=a\cdot\left(  -1\right)  \in R_{m}$. This
proves Claim 3.]

Now, consider the set $R_{m}$. We have $0\in R_{m}$ (by Claim 1, applied to
$r=0$) and $1\in R_{m}$ (by Claim 1, applied to $r=1$). Thus, the set $R_{m}$
contains the elements $0$ and $1$.

Furthermore, every $a\in R_{m}$ and $b\in R_{m}$ satisfy $a+b\in R_{m}$ (by
Claim 2). Thus, we can define a binary operation $+$ on the set $R_{m}$ by
restricting the usual addition $+$ on $\mathbb{Q}$ to the subset $R_{m}$.

Moreover, every $a\in R_{m}$ and $b\in R_{m}$ satisfy $a\cdot b\in R_{m}$ (by
Claim 2). Thus, we can define a binary operation $\cdot$ on the set $R_{m}$ by
restricting the usual multiplication $\cdot$ on $\mathbb{Q}$ to the subset
$R_{m}$.

Now, the exercise demands us to prove that the set $R_{m}$ equipped with these
two binary operations $+$ and $\cdot$ and the two elements $0$ and $1$ is a
commutative ring. In order to do so, we must verify that the ring axioms and
the \textquotedblleft Commutativity of multiplication\textquotedblright\ axiom
are satisfied.

But all of these axioms, except for the \textquotedblleft Existence of
additive inverses\textquotedblright\ axiom, are clearly satisfied because they
are satisfied for $\mathbb{Q}$ (and because our $R_{m}$ is a subset of
$\mathbb{Q}$, and because we endowed $R_{m}$ with operations $+$ and $\cdot$
that are restrictions of the corresponding operations of $\mathbb{Q}$). It
thus remains to prove that the \textquotedblleft Existence of additive
inverses\textquotedblright\ axiom is satisfied.

But this is easy: If $a\in R_{m}$, then $-a\in R_{m}$ (by Claim 3), and thus
there exists an element $a^{\prime}\in R_{m}$ satisfying $a+a^{\prime
}=a^{\prime}+a=0$ (namely, $a^{\prime}=-a$).

Thus, we have proven that $R_{m}$ is a commutative ring. This solves part
\textbf{(a)} of the exercise.

\bigskip

\textbf{(b)} From (\ref{sol.ring.m-int.equiv}) (applied to $r=x$), we obtain
the logical equivalence
\[
\left(  x\in R_{m}\right)  \ \Longleftrightarrow\ \left(  \text{there exists a
}k\in\mathbb{N}\text{ satisfying }m^{k}x\in\mathbb{Z}\right)  .
\]
But Exercise 3.4.2 \textbf{(d)} in the
\href{http://www.cip.ifi.lmu.de/~grinberg/t/19s/notes.pdf}{class notes}
(applied to $r=x$) yields the logical equivalence%
\begin{align*}
&  \ \left(  \text{there exists a }k\in\mathbb{N}\text{ satisfying }m^{k}%
x\in\mathbb{Z}\right) \\
&  \Longleftrightarrow\ \left(  \text{every prime }p\text{ satisfying }%
w_{p}\left(  x\right)  <0\text{ satisfies }p\mid m\right)  .
\end{align*}
Hence, we have the following chain of equivalences:%
\begin{align*}
\left(  x\in R_{m}\right)  \  &  \Longleftrightarrow\ \left(  \text{there
exists a }k\in\mathbb{N}\text{ satisfying }m^{k}x\in\mathbb{Z}\right) \\
&  \Longleftrightarrow\ \left(  \text{every prime }p\text{ satisfying }%
w_{p}\left(  x\right)  <0\text{ satisfies }p\mid m\right)  .
\end{align*}
This solves part \textbf{(b)} of the exercise.

%----------------------------------------------------------------------------------------
%	EXERCISE 2
%----------------------------------------------------------------------------------------
\rule{\linewidth}{0.3pt} \\[0.4cm]

\section{Exercise 2: Rings with $x^{2} = x$}

\subsection{Problem}

Let $\mathbb{K}$ be a ring with the property that
\begin{equation}
u^{2}=u\qquad\text{ for all }u\in\mathbb{K}. \label{eq.exe.ring.xx=x.cond}%
\end{equation}

(Examples of such rings are $\mathbb{Z}/2$ as well as the \textquotedblleft
power set\textquotedblright\ ring $\left(  \mathcal{P}\left(  S\right)
,\triangle,\cap,\varnothing,S\right)  $ constructed from any given set $S$.)

Prove the following:

\begin{enumerate}
\item[\textbf{(a)}] We have $2x = 0$ for each $x \in\mathbb{K}$.

\item[\textbf{(b)}] We have $-x = x$ for each $x \in\mathbb{K}$.

\item[\textbf{(c)}] We have $xy = yx$ for all $x, y \in\mathbb{K}$. (In other
words, the ring $\mathbb{K}$ is commutative.)
\end{enumerate}

(As usual, ``$0$'' stands for the zero of the ring $\mathbb{K}$.)

[\textbf{Hint:} For part \textbf{(a)}, apply \eqref{eq.exe.ring.xx=x.cond} to
$u=x$ but also to $u=2x=x+x$, and see what comes out. For part \textbf{(c)},
apply \eqref{eq.exe.ring.xx=x.cond} to $u=x+y$.]

\subsection{Remark}

Rings $\mathbb{K}$ satisfying \eqref{eq.exe.ring.xx=x.cond} are known as
\textit{Boolean rings} (although some people do not require them to have a
unity). Thus, the exercise proves various properties of Boolean rings,
including the fact that they are always commutative.

You might wonder what happens if we replace \eqref{eq.exe.ring.xx=x.cond} by
the requirement that
\begin{equation}
u^{3}=u\qquad\text{ for all }u\in\mathbb{K}. \label{eq.exe.ring.xx=x.cond3}%
\end{equation}
This no longer leads to $2x=0$ (nor to $3x=0$ as you might perhaps expect).
Instead, it can be shown that $6x=0$ for all $x\in\mathbb{K}$. It can also be
shown that it leads to $xy=yx$. See, for example,
\url{https://math.stackexchange.com/questions/67148} .

More generally, fix an integer $n\geq2$, and replace
\eqref{eq.exe.ring.xx=x.cond} by the requirement that
\begin{equation}
u^{n}=u\qquad\text{ for all }u\in\mathbb{K}. \label{eq.exe.ring.xx=x.condn}%
\end{equation}
Then, it still can be proven that $\mathbb{K}$ is commutative! See
\url{https://mathoverflow.net/questions/29590/} for this result.

Even more generally, we don't need to fix $n$ in advance! In other words,
instead of requiring \eqref{eq.exe.ring.xx=x.cond} or
\eqref{eq.exe.ring.xx=x.cond3} or \eqref{eq.exe.ring.xx=x.condn}, we merely
require that for each $u\in\mathbb{K}$, there exists an integer $n\geq2$
(which may depend on $u$) such that $u^{n}=u$. This is a more general setting;
nevertheless it still follows that $\mathbb{K}$ is commutative! This is a
result of Jacobson; see \cite{Rogers71} for a proof.

\subsection{Solution}

\textbf{(a)} \textit{First solution to part \textbf{(a)}:} Let $x\in
\mathbb{K}$. Then, \eqref{eq.exe.ring.xx=x.cond} (applied to $u=x$) yields
$x^{2}=x$. But \eqref{eq.exe.ring.xx=x.cond} (applied to $u=x+x$) yields
$\left(  x+x\right)  ^{2}=x+x=2x$. Thus,%
\begin{align*}
2x  &  =x+x=\left(  x+x\right)  ^{2}=\left(  x+x\right)  \left(  x+x\right) \\
&  =\underbrace{x\left(  x+x\right)  }_{\substack{=xx+xx\\\text{(by
distributivity)}}}+\underbrace{x\left(  x+x\right)  }%
_{\substack{=xx+xx\\\text{(by distributivity)}}}\qquad\left(  \text{by
distributivity}\right) \\
&  =xx+xx+xx+xx=4\underbrace{xx}_{=x^{2}=x}=4x.
\end{align*}
Subtracting $2x$ from both sides of this equality, we obtain $0=4x-2x=2x$.
Hence, $2x=0$. This solves part \textbf{(a)} of the exercise.

\textit{Second solution to part \textbf{(a)}:} Let $x\in\mathbb{K}$. Then,
\eqref{eq.exe.ring.xx=x.cond} (applied to $u=x$) yields $x^{2}=x$. But
\eqref{eq.exe.ring.xx=x.cond} (applied to $u=-x$) yields $\left(  -x\right)
^{2}=-x$. Hence,%
\[
-x=\left(  -x\right)  ^{2}=\left(  -x\right)  \left(  -x\right)
=-\underbrace{\left(  x\cdot\left(  -x\right)  \right)  }_{=-\left(
xx\right)  }=-\left(  -\left(  xx\right)  \right)  =xx=x^{2}=x.
\]
Adding $x$ to both sides of this equality, we find $\left(  -x\right)
+x=x+x=2x$. Thus, $2x=\left(  -x\right)  +x=0$. This solves part \textbf{(a)}
of the exercise.

\bigskip

\textbf{(b)} \textit{First solution to part \textbf{(b)}:} Let $x\in
\mathbb{K}$. Part \textbf{(a)} of this exercise yields $2x=0$. Subtracting $x$
from both sides of this equality, we obtain $2x-x=0-x=-x$. Hence,
$-x=\underbrace{2x}_{=x+x}-x=\left(  x+x\right)  -x=x$. This solves part
\textbf{(b)} of this exercise.

\textit{Second solution to part \textbf{(b)}:} Let $x\in\mathbb{K}$. We have
already shown the equality $-x=x$ in our Second solution to part \textbf{(a)}.
Thus, part \textbf{(b)} of the exercise is solved.

\bigskip

\textbf{(c)} Let $x,y\in\mathbb{K}$. Then, \eqref{eq.exe.ring.xx=x.cond}
(applied to $u=x$) yields $x^{2}=x$. Also, \eqref{eq.exe.ring.xx=x.cond}
(applied to $u=y$) yields $y^{2}=y$. But \eqref{eq.exe.ring.xx=x.cond}
(applied to $u=x+y$) yields $\left(  x+y\right)  ^{2}=x+y$. Hence,%
\begin{align*}
x+y  &  =\left(  x+y\right)  ^{2}=\left(  x+y\right)  \left(  x+y\right) \\
&  =\underbrace{x\left(  x+y\right)  }_{\substack{=xx+xy\\\text{(by
distributivity)}}}+\underbrace{y\left(  x+y\right)  }%
_{\substack{=yx+yy\\\text{(by distributivity)}}}\qquad\left(  \text{by
distributivity}\right) \\
&  =\underbrace{xx}_{=x^{2}=x}+xy+yx+\underbrace{yy}_{=y^{2}=y}=x+xy+yx+y.
\end{align*}
Subtracting $x+y$ from both sides of this equality, we obtain%
\[
0=\left(  x+xy+yx+y\right)  -\left(  x+y\right)  =xy+yx.
\]
Thus, $xy=-yx$. But part \textbf{(b)} of this exercise (applied to $yx$
instead of $x$) yields $-yx=yx$. Hence, $xy=-yx=yx$. This solves part
\textbf{(c)} of the exercise.

%----------------------------------------------------------------------------------------
%	EXERCISE 3
%----------------------------------------------------------------------------------------
\rule{\linewidth}{0.3pt} \\[0.4cm]

\section{Exercise 3: A matrix of gcds}

\subsection{Problem}

In this exercise, we shall again use
\href{https://en.wikipedia.org/wiki/Iverson_bracket}{the \textit{Iverson
bracket notation}}:

Let $n \in\mathbb{N}$. Let $G$ be the $n \times n$-matrix
\[
\left(  \gcd\left(  i, j \right)  \right)  _{1 \leq i \leq n, \ 1 \leq j \leq
n } =
\begin{pmatrix}
\gcd\left(  1, 1 \right)  & \gcd\left(  1, 2 \right)  & \cdots & \gcd\left(
1, n \right) \\
\gcd\left(  2, 1 \right)  & \gcd\left(  2, 2 \right)  & \cdots & \gcd\left(
2, n \right) \\
\vdots & \vdots & \ddots & \vdots\\
\gcd\left(  n, 1 \right)  & \gcd\left(  n, 2 \right)  & \cdots & \gcd\left(
n, n \right)
\end{pmatrix}
.
\]

Let $L$ be the $n \times n$-matrix
\[
\left(  \left[  j \mid i \right]  \right)  _{1 \leq i \leq n, \ 1 \leq j \leq
n } =
\begin{pmatrix}
\left[  1 \mid1 \right]  & \left[  2 \mid1 \right]  & \cdots & \left[  n \mid1
\right] \\
\left[  1 \mid2 \right]  & \left[  2 \mid2 \right]  & \cdots & \left[  n \mid2
\right] \\
\vdots & \vdots & \ddots & \vdots\\
\left[  1 \mid n \right]  & \left[  2 \mid n \right]  & \cdots & \left[  n
\mid n \right]
\end{pmatrix}
.
\]

Let $D$ be the $n \times n$-matrix
\[
\left(  \left[  i = j \right]  \phi\left(  i \right)  \right)  _{1 \leq i \leq
n, \ 1 \leq j \leq n } =
\begin{pmatrix}
\phi\left(  1 \right)  & 0 & 0 & \cdots & 0\\
0 & \phi\left(  2 \right)  & 0 & \cdots & 0\\
0 & 0 & \phi\left(  3 \right)  & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & \cdots & \phi\left(  n \right)
\end{pmatrix}
.
\]

Prove that\footnote{We are using the standard notation $A^{T}$ for the
\textit{transpose} of a matrix $A$. This transpose is defined as follows: If
$A = \left(  a_{i, j} \right)  _{1 \leq i \leq n, \ 1 \leq j \leq m }$, then
$A^{T} = \left(  a_{j, i} \right)  _{1 \leq i \leq m, \ 1 \leq j \leq n }$.}
$G = LDL^{T}$.

\subsection{Remark}

As the names suggest, the matrix $L$ is lower-triangular\footnote{because two
positive integers $i$ and $j$ satisfying $i<j$ always satisfy $\left[  j\mid
i\right]  =0$} (so that the matrix $L^{T}$ is upper-triangular), and the
matrix $D$ is diagonal. Thus, $G=LDL^{T}$ is an instance of an
\href{https://en.wikipedia.org/wiki/LU_decomposition#LDU_decomposition}{\textit{LDU
decomposition}}.

The matrix $G$ is an example of what is called a \textit{gcd-matrix} (duh);
see, e.g., \url{https://math.stackexchange.com/questions/1278871} for some
references on it.

\subsection{Solution}

We recall the following fundamental property of matrices in general:

\begin{lemma}
\label{lem.sol.matrix.gcd-LDU.row-mult}Let $n,m\in\mathbb{N}$. Let
$\mathbb{K}$ be a ring. Let $\mathbf{D}=\left(  \left[  i=j\right]
d_{i}\right)  _{1\leq i\leq n,\ 1\leq j\leq n}$ be a diagonal matrix over
$\mathbb{K}$ (where $d_{1},d_{2},\ldots,d_{n}$ are $n$ elements of
$\mathbb{K}$). Let $\mathbf{A}=\left(  a_{i,j}\right)  _{1\leq i\leq n,\ 1\leq
j\leq m}$ be an arbitrary $n\times m$-matrix over $\mathbb{K}$. Then, the
product $\mathbf{D}\mathbf{A}$ is simply given by
\begin{equation}
\mathbf{D}\mathbf{A}=\left(  d_{i}a_{i,j}\right)  _{1\leq i\leq n,\ 1\leq
j\leq m}. \label{eq.matrix.gcd-LDU.row-mult}%
\end{equation}
(That is, multiplying a matrix $\mathbf{A}$ by a diagonal matrix $\mathbf{D}$
on the left is tantamount to rescaling each row of $\mathbf{A}$ by the
corresponding diagonal entry of $\mathbf{D}$.)
\end{lemma}

\begin{proof}
[Proof of Lemma \ref{lem.sol.matrix.gcd-LDU.row-mult}.]We have $\mathbf{D}%
=\left(  \left[  i=j\right]  d_{i}\right)  _{1\leq i\leq n,\ 1\leq j\leq n}$
and $\mathbf{A}=\left(  a_{i,j}\right)  _{1\leq i\leq n,\ 1\leq j\leq m}$.
Hence, the definition of the product of two matrices yields%
\begin{equation}
\mathbf{DA}=\left(  \sum_{k=1}^{n}\left[  i=k\right]  d_{i}a_{k,j}\right)
_{1\leq i\leq n,\ 1\leq j\leq m}. \label{pf.lem.sol.matrix.gcd-LDU.row-mult.1}%
\end{equation}
Now, fix some $i\in\left\{  1,2,\ldots,n\right\}  $ and $j\in\left\{
1,2,\ldots,m\right\}  $. Then,%
\begin{align*}
\sum_{k=1}^{n}\left[  i=k\right]  d_{i}a_{k,j}  &  =\sum_{k\in\left\{
1,2,\ldots,n\right\}  }\left[  i=k\right]  d_{i}a_{k,j}=\underbrace{\left[
i=i\right]  }_{\substack{=1\\\text{(since }i=i\text{)}}}d_{i}a_{i,j}%
+\sum_{\substack{k\in\left\{  1,2,\ldots,n\right\}  ;\\k\neq i}%
}\underbrace{\left[  i=k\right]  }_{\substack{=0\\\text{(since }i\neq
k\\\text{(because }k\neq i\text{))}}}d_{i}a_{k,j}\\
&  \qquad\left(  \text{here, we have split off the addend for }k=i\text{ from
the sum}\right) \\
&  =d_{i}a_{i,j}+\underbrace{\sum_{\substack{k\in\left\{  1,2,\ldots
,n\right\}  ;\\k\neq i}}0d_{i}a_{k,j}}_{=0}=d_{i}a_{i,j}.
\end{align*}
Now, forget that we fixed $i$ and $j$. We thus have shown that $\sum_{k=1}%
^{n}\left[  i=k\right]  d_{i}a_{k,j}=d_{i}a_{i,j}$ for all $i\in\left\{
1,2,\ldots,n\right\}  $ and $j\in\left\{  1,2,\ldots,m\right\}  $. Therefore,
the equality \eqref{pf.lem.sol.matrix.gcd-LDU.row-mult.1} rewrites as
$\mathbf{D}\mathbf{A}=\left(  d_{i}a_{i,j}\right)  _{1\leq i\leq n,\ 1\leq
j\leq m}$. This proves Lemma \ref{lem.sol.matrix.gcd-LDU.row-mult}.
\end{proof}

Now, let us return to solving the exercise.

Recall that $L=\left(  \left[  j\mid i\right]  \right)  _{1\leq i\leq
n,\ 1\leq j\leq n}$; thus, the definition of the transpose of a matrix yields
$L^{T}=\left(  \left[  i\mid j\right]  \right)  _{1\leq i\leq n,\ 1\leq j\leq
n}$. Also, recall that $D=\left(  \left[  i=j\right]  \phi\left(  i\right)
\right)  _{1\leq i\leq n,\ 1\leq j\leq n}$. Hence,
\eqref{eq.matrix.gcd-LDU.row-mult} (applied to $\mathbb{K}=\mathbb{Z}$, $m=n$,
$d_{i}=\phi\left(  i\right)  $, $\mathbf{D}=D$, $a_{i,j}=\left[  i\mid
j\right]  $ and $\mathbf{A}=L^{T}$) yields%
\begin{equation}
DL^{T}=\left(  \phi\left(  i\right)  \cdot\left[  i\mid j\right]  \right)
_{1\leq i\leq n,\ 1\leq j\leq n}. \label{sol.matrix.gcd-LDU.1}%
\end{equation}

Now, we have $L=\left(  \left[  j\mid i\right]  \right)  _{1\leq i\leq
n,\ 1\leq j\leq n}$ and $DL^{T}=\left(  \phi\left(  i\right)  \cdot\left[
i\mid j\right]  \right)  _{1\leq i\leq n,\ 1\leq j\leq n}$. Thus, the
definition of the product of two matrices yields%
\begin{equation}
L\left(  DL^{T}\right)  =\left(  \sum_{k=1}^{n}\left[  k\mid i\right]
\cdot\phi\left(  k\right)  \cdot\left[  k\mid j\right]  \right)  _{1\leq i\leq
n,\ 1\leq j\leq n}. \label{sol.matrix.gcd-LDU.2}%
\end{equation}

Let us now recall a basic property of the Iverson bracket: If $\mathcal{A}$
and $\mathcal{B}$ are two logical statements, then
\begin{equation}
\left[  \mathcal{A}\wedge\mathcal{B}\right]  =\left[  \mathcal{A}\right]
\cdot\left[  \mathcal{B}\right]  . \label{sol.matrix.gcd-LDU.iverAB}%
\end{equation}
Furthermore, if $\mathcal{A}$ and $\mathcal{B}$ are two equivalent logical
statements, then%
\begin{equation}
\left[  \mathcal{A}\right]  =\left[  \mathcal{B}\right]  .
\label{sol.matrix.gcd-LDU.iverA=B}%
\end{equation}

Now, fix $i,j\in\left\{  1,2,\ldots,n\right\}  $. Hence, $i$ and $j$ are
positive integers; thus, $\gcd\left(  i,j\right)  $ is a positive integer as
well. Also, $\gcd\left(  i,j\right)  \mid i$, so that $\gcd\left(  i,j\right)
\leq i$ (since $\gcd\left(  i,j\right)  $ and $i$ are positive integers) and
therefore $\gcd\left(  i,j\right)  \leq i\leq n$ (since $i\in\left\{
1,2,\ldots,n\right\}  $).

Theorem 2.14.6 in the
\href{http://www.cip.ifi.lmu.de/~grinberg/t/19s/notes.pdf}{class notes}
(applied to $\gcd\left(  i,j\right)  $ instead of $n$) yields%
\begin{equation}
\sum_{d\mid\gcd\left(  i,j\right)  }\phi\left(  d\right)  =\gcd\left(
i,j\right)  . \label{sol.matrix.gcd-LDU.sum-phi}%
\end{equation}
Here, the summation sign \textquotedblleft$\sum_{d\mid\gcd\left(  i,j\right)
}$\textquotedblright\ means a sum over all positive divisors $d$ of
$\gcd\left(  i,j\right)  $.

Also, fix $k\in\left\{  1,2,\ldots,n\right\}  $. Then, Theorem 2.9.15
\textbf{(a)} in the
\href{http://www.cip.ifi.lmu.de/~grinberg/t/19s/notes.pdf}{class notes}
(applied to $k$, $i$ and $j$ instead of $a$, $b$ and $m$) shows that we have
the following logical equivalence:%
\[
\left(  k\mid i\text{ and }k\mid j\right)  \Longleftrightarrow\left(
k\mid\gcd\left(  i,j\right)  \right)  .
\]
Thus, the logical statements $\left(  k\mid\gcd\left(  i,j\right)  \right)  $
and $\left(  k\mid i\text{ and }k\mid j\right)  $ are equivalent. Hence,
\eqref{sol.matrix.gcd-LDU.iverA=B} (applied to $\mathcal{A}=\left(  k\mid
\gcd\left(  i,j\right)  \right)  $ and $\mathcal{B}=\left(  k\mid i\text{ and
}k\mid j\right)  $) yields%
\begin{align}
\left[  k\mid\gcd\left(  i,j\right)  \right]   &  =\left[  k\mid i\text{ and
}k\mid j\right]  =\left[  \left(  k\mid i\right)  \wedge\left(  k\mid
j\right)  \right] \nonumber\\
&  =\left[  k\mid i\right]  \cdot\left[  k\mid j\right]
\label{sol.matrix.gcd-LDU.3}%
\end{align}
(by \eqref{sol.matrix.gcd-LDU.iverAB}, applied to $\mathcal{A}=\left(  k\mid
i\right)  $ and $\mathcal{B}=\left(  k\mid j\right)  $).

Now, forget that we fixed $k$. We thus have proven
\eqref{sol.matrix.gcd-LDU.3} for each $k\in\left\{  1,2,\ldots,n\right\}  $.
Now,%
\begin{align}
&  \sum_{k=1}^{n}\left[  k\mid i\right]  \cdot\phi\left(  k\right)
\cdot\left[  k\mid j\right] \nonumber\\
&  =\underbrace{\sum_{k=1}^{n}}_{=\sum_{k\in\left\{  1,2,\ldots,n\right\}  }%
}\underbrace{\left[  k\mid i\right]  \cdot\left[  k\mid j\right]
}_{\substack{=\left[  k\mid\gcd\left(  i,j\right)  \right]  \\\text{(by
\eqref{sol.matrix.gcd-LDU.3})}}}\cdot\phi\left(  k\right)  =\sum_{k\in\left\{
1,2,\ldots,n\right\}  }\left[  k\mid\gcd\left(  i,j\right)  \right]  \cdot
\phi\left(  k\right) \nonumber\\
&  =\sum_{\substack{k\in\left\{  1,2,\ldots,n\right\}  ;\\k\mid\gcd\left(
i,j\right)  }}\underbrace{\left[  k\mid\gcd\left(  i,j\right)  \right]
}_{\substack{=1\\\text{(since }k\mid\gcd\left(  i,j\right)  \text{)}}%
}\cdot\phi\left(  k\right)  +\sum_{\substack{k\in\left\{  1,2,\ldots
,n\right\}  ;\\k\nmid\gcd\left(  i,j\right)  }}\underbrace{\left[  k\mid
\gcd\left(  i,j\right)  \right]  }_{\substack{=0\\\text{(since }k\nmid
\gcd\left(  i,j\right)  \text{)}}}\cdot\phi\left(  k\right) \nonumber\\
&  \qquad\left(
\begin{array}
[c]{c}%
\text{since each }k\in\left\{  1,2,\ldots,n\right\}  \text{ satisfies either
}k\mid\gcd\left(  i,j\right) \\
\text{or }k\nmid\gcd\left(  i,j\right)  \text{ (but not both)}%
\end{array}
\right) \nonumber\\
&  =\sum_{\substack{k\in\left\{  1,2,\ldots,n\right\}  ;\\k\mid\gcd\left(
i,j\right)  }}\phi\left(  k\right)  +\underbrace{\sum_{\substack{k\in\left\{
1,2,\ldots,n\right\}  ;\\k\nmid\gcd\left(  i,j\right)  }}0\cdot\phi\left(
k\right)  }_{=0}=\sum_{\substack{k\in\left\{  1,2,\ldots,n\right\}
;\\k\mid\gcd\left(  i,j\right)  }}\phi\left(  k\right) \nonumber\\
&  =\sum_{\substack{d\in\left\{  1,2,\ldots,n\right\}  ;\\d\mid\gcd\left(
i,j\right)  }}\phi\left(  d\right)  \label{sol.matrix.gcd-LDU.5}%
\end{align}
(here, we have renamed the summation index $k$ as $d$).

But each positive divisor of $\gcd\left(  i,j\right)  $ is $\leq\gcd\left(
i,j\right)  $ and therefore $\leq n$ (since $\gcd\left(  i,j\right)  \leq n$).
Thus, each positive divisor of $\gcd\left(  i,j\right)  $ belongs to the set
$\left\{  1,2,\ldots,n\right\}  $. Hence, each positive divisor of
$\gcd\left(  i,j\right)  $ is a $d\in\left\{  1,2,\ldots,n\right\}  $
satisfying $d\mid\gcd\left(  i,j\right)  $. In other words,%
\[
\left\{  \text{positive divisors of }\gcd\left(  i,j\right)  \right\}
\subseteq\left\{  d\in\left\{  1,2,\ldots,n\right\}  \text{ such that }%
d\mid\gcd\left(  i,j\right)  \right\}  .
\]
Combining this with the relation%
\[
\left\{  d\in\left\{  1,2,\ldots,n\right\}  \text{ such that }d\mid\gcd\left(
i,j\right)  \right\}  \subseteq\left\{  \text{positive divisors of }%
\gcd\left(  i,j\right)  \right\}
\]
(which is obvious), we obtain%
\[
\left\{  d\in\left\{  1,2,\ldots,n\right\}  \text{ such that }d\mid\gcd\left(
i,j\right)  \right\}  =\left\{  \text{positive divisors of }\gcd\left(
i,j\right)  \right\}  .
\]
Thus, the summation sign \textquotedblleft$\sum_{\substack{d\in\left\{
1,2,\ldots,n\right\}  ;\\d\mid\gcd\left(  i,j\right)  }}$\textquotedblright%
\ is equivalent to \textquotedblleft$\sum_{d\mid\gcd\left(  i,j\right)  }%
$\textquotedblright\ (since the latter summation sign means a sum over all
positive divisors of $\gcd\left(  i,j\right)  $). Hence,
\eqref{sol.matrix.gcd-LDU.5} becomes%
\[
\sum_{k=1}^{n}\left[  k\mid i\right]  \cdot\phi\left(  k\right)  \cdot\left[
k\mid j\right]  =\underbrace{\sum_{\substack{d\in\left\{  1,2,\ldots
,n\right\}  ;\\d\mid\gcd\left(  i,j\right)  }}}_{=\sum_{d\mid\gcd\left(
i,j\right)  }}\phi\left(  d\right)  =\sum_{d\mid\gcd\left(  i,j\right)  }%
\phi\left(  d\right)  =\gcd\left(  i,j\right)  \qquad\qquad\left(  \text{by
\eqref{sol.matrix.gcd-LDU.sum-phi}}\right)  .
\]

Now, forget that we fixed $i,j$. We thus have proven that
\[
\sum_{k=1}^{n}\left[  k\mid i\right]  \cdot\phi\left(  k\right)  \cdot\left[
k\mid j\right]  =\gcd\left(  i,j\right)  \qquad\text{for all }i,j\in\left\{
1,2,\ldots,n\right\}  .
\]
In other words,%
\[
\left(  \sum_{k=1}^{n}\left[  k\mid i\right]  \cdot\phi\left(  k\right)
\cdot\left[  k\mid j\right]  \right)  _{1\leq i\leq n,\ 1\leq j\leq n}=\left(
\gcd\left(  i,j\right)  \right)  _{1\leq i\leq n,\ 1\leq j\leq n}.
\]
Hence, \eqref{sol.matrix.gcd-LDU.2} becomes%
\[
L\left(  DL^{T}\right)  =\left(  \sum_{k=1}^{n}\left[  k\mid i\right]
\cdot\phi\left(  k\right)  \cdot\left[  k\mid j\right]  \right)  _{1\leq i\leq
n,\ 1\leq j\leq n}=\left(  \gcd\left(  i,j\right)  \right)  _{1\leq i\leq
n,\ 1\leq j\leq n}=G
\]
(by the definition of $G$). Consequently, $G=L\left(  DL^{T}\right)  =LDL^{T}%
$. The exercise is now solved.

\subsection{Remark}

If you know about determinants, you will easily see how to compute $\det G$
using the claim of the exercise. (\textbf{Hint:} Use the formula $\det\left(
AB\right)  =\det A\cdot\det B$ that holds for any two $n\times n$-matrices $A$
and $B$ over any commutative ring $\mathbb{K}$.)

%----------------------------------------------------------------------------------------
%	EXERCISE 4
%----------------------------------------------------------------------------------------
\rule{\linewidth}{0.3pt} \\[0.4cm]

\section{Exercise 4: Idempotent and involutive elements}

\subsection{Problem}

Let $\mathbb{K}$ be a ring.

An element $a$ of $\mathbb{K}$ is said to be \textit{idempotent} if it
satisfies $a^{2} = a$.

An element $a$ of $\mathbb{K}$ is said to be \textit{involutive} if it
satisfies $a^{2} = 1$.

\begin{enumerate}
\item[\textbf{(a)}] Let $a \in\mathbb{K}$. Prove that if $a$ is idempotent,
then $1 - 2a$ is involutive.

\item[\textbf{(b)}] Now, assume that $2$ is \textit{cancellable} in
$\mathbb{K}$; this means that if $u$ and $v$ are two elements of $\mathbb{K}$
satisfying $2u=2v$, then $u=v$. Prove that the converse of the claim of part
\textbf{(a)} holds: If $a\in\mathbb{K}$ is such that $1-2a$ is involutive,
then $a$ is idempotent.

\item[\textbf{(c)}] Now, let $\mathbb{K}=\mathbb{Z}/4$. Find an element
$a\in\mathbb{K}$ such that $1-2a$ is involutive, but $a$ is not idempotent.
\end{enumerate}

\subsection{Remark}

The idempotent elements of $\mathbb{R}$ are $0$ and $1$. The involutive
elements of $\mathbb{R}$ are $1$ and $-1$. A matrix ring like $\mathbb{R}%
^{n\times n}$ usually has infinitely many idempotent elements (viz., all
projection matrices on subspaces of $\mathbb{R}^{n}$) and infinitely many
involutive elements (viz., all matrices $A$ satisfying $A^{2}=I_{n}$; for
instance, all reflections across hyperplanes are represented by such matrices).

Part \textbf{(a)} of this exercise assigns an involutive element to each
idempotent element of $\mathbb{K}$. If $2$ is invertible in $\mathbb{K}$ (that
is, if the element $2\cdot1_{\mathbb{K}}$ has a multiplicative inverse), then
this assignment is a bijection (as can be easily derived from part
\textbf{(b)}). Part \textbf{(c)} shows that we cannot drop the
\textquotedblleft$2$ is cancellable\textquotedblright\ condition in part
\textbf{(b)}.

\subsection{Solution}

Using the ring axioms and the basic rules for rings, it is easy to see that
every $a\in\mathbb{K}$ satisfies%
\begin{equation}
\left(  1-2a\right)  ^{2}=1-4a+4a^{2}. \label{sol.ring.idp-inv.1}%
\end{equation}

[Here is a more pedantic \textit{proof} of this fact: Let $a\in\mathbb{K}$.
For any $b\in\mathbb{K}$, we have%
\begin{align*}
\left(  1+b\right)  ^{2}  &  =\left(  1+b\right)  \left(  1+b\right)
=\underbrace{1\left(  1+b\right)  }_{=1+b}+\underbrace{b\left(  1+b\right)
}_{\substack{=b\cdot1+b\cdot b\\\text{(by distributivity)}}}\qquad\left(
\text{by distributivity}\right) \\
&  =1+b+\underbrace{b\cdot1}_{=b}+\underbrace{b\cdot b}_{=b^{2}}%
=1+b+b+b^{2}=1+2b+b^{2}.
\end{align*}
Applying this to $b=-2a$, we obtain%
\begin{align*}
\left(  1+\left(  -2a\right)  \right)  ^{2}  &  =1+\underbrace{2\left(
-2a\right)  }_{=2\cdot\left(  -2\right)  a}+\underbrace{\left(  -2a\right)
^{2}}_{\substack{=\left(  -2a\right)  \cdot\left(  -2a\right)  \\=\left(
-2\right)  \cdot\left(  -2a\right)  \cdot a}}=1+\underbrace{2\cdot\left(
-2\right)  }_{=-4}a+\left(  -2\right)  \cdot\underbrace{\left(  -2a\right)
}_{=\left(  -2\right)  a}\cdot a\\
&  =1+\underbrace{\left(  -4\right)  a}_{=-4a}+\underbrace{\left(  -2\right)
\cdot\left(  \left(  -2\right)  a\right)  }_{=\left(  \left(  -2\right)
\cdot\left(  -2\right)  \right)  \cdot a}\cdot a=\underbrace{1+\left(
-4a\right)  }_{=1-4a}+\underbrace{\left(  \left(  -2\right)  \cdot\left(
-2\right)  \right)  }_{=4}\cdot\underbrace{a\cdot a}_{=a^{2}}\\
&  =1-4a+4a^{2}.
\end{align*}
In view of $1+\left(  -2a\right)  =1-2a$, this rewrites as $\left(
1-2a\right)  ^{2}=1-4a+4a^{2}$. Thus, \eqref{sol.ring.idp-inv.1} is proven.]

\bigskip

\textbf{(a)} Assume that $a$ is idempotent. We must prove that $1-2a$ is involutive.

We have assumed that $a$ is idempotent. In other words, $a^{2}=a$ (by the
definition of \textquotedblleft idempotent\textquotedblright). Thus,
\eqref{sol.ring.idp-inv.1} becomes $\left(  1-2a\right)  ^{2}%
=1-4a+4\underbrace{a^{2}}_{=a}=1-4a+4a=1$. In other words, $1-2a$ is
involutive (by the definition of \textquotedblleft
involutive\textquotedblright). This solves part \textbf{(a)} of the exercise.

\bigskip

\textbf{(b)} Let $a\in\mathbb{K}$ be such that $1-2a$ is involutive. We must
prove that $a$ is idempotent.

We know that $2$ is cancellable in $\mathbb{K}$. In other words, if $u$ and
$v$ are two elements of $\mathbb{K}$ satisfying $2u=2v$, then
\begin{equation}
u=v. \label{sol.ring.idp-inv.b.u=v}%
\end{equation}

We have assumed that $1-2a$ is involutive. In other words, $\left(
1-2a\right)  ^{2}=1$. Comparing this with \eqref{sol.ring.idp-inv.1}, we
obtain $1-4a+4a^{2}=1$. Hence, $4a^{2}=1-1+4a=4a$. This rewrites as
$2\cdot2a^{2}=2\cdot2a$ (since $\underbrace{2\cdot2}_{=4}a^{2}=4a^{2}$ and
$\underbrace{2\cdot2}_{=4}a=4a$). Hence, \eqref{sol.ring.idp-inv.b.u=v}
(applied to $u=2a^{2}$ and $v=2a$) yields $2a^{2}=2a$. Thus,
\eqref{sol.ring.idp-inv.b.u=v} (applied to $u=a^{2}$ and $v=a$) yields
$a^{2}=a$. In other words, $a$ is idempotent. This solves part \textbf{(b)} of
the exercise.

\bigskip

\textbf{(c)} We claim that $a=\left[  2\right]  _{4}$ is such an element.
Indeed, $1-2\cdot\left[  2\right]  _{4}$ is involutive\footnote{Keep in mind
that the \textquotedblleft$1$\textquotedblright\ here stands for the unity of
the ring $\mathbb{K}=\mathbb{Z}/4$; this is the residue class $\left[
1\right]  _{4}$.} (since $\underbrace{1}_{=\left[  1\right]  _{4}%
}-\underbrace{2\cdot\left[  2\right]  _{4}}_{=\left[  2\cdot2\right]
_{4}=\left[  4\right]  _{4}=\left[  0\right]  _{4}}=\left[  1\right]
_{4}-\left[  0\right]  _{4}=\left[  1\right]  _{4}$ and thus $\left(
1-2\cdot\left[  2\right]  _{4}\right)  ^{2}=\left(  \left[  1\right]
_{4}\right)  ^{2}=\left[  1^{2}\right]  _{4}=\left[  1\right]  _{4}=1$), but
$\left[  2\right]  _{4}$ is not idempotent (since $\left(  \left[  2\right]
_{4}\right)  ^{2}=\left[  2^{2}\right]  _{4}=\left[  4\right]  _{4}\neq\left[
2\right]  _{4}$).

%----------------------------------------------------------------------------------------
%	EXERCISE 5
%----------------------------------------------------------------------------------------
\rule{\linewidth}{0.3pt} \\[0.4cm]

\section{Exercise 5: The matrix approach to Fibonacci numbers}

\subsection{Problem}

Let $A$ be the $2 \times2$-matrix $%
\begin{pmatrix}
0 & 1\\
1 & 1
\end{pmatrix}
$ over $\mathbb{Z}$. Consider also the identity matrix $I_{2} \in\mathbb{Z}^{2
\times2}$.

Let $\mathcal{F}$ be the subset
\[
\left\{  aA+bI_{2}\mid a,b\in\mathbb{Z}\right\}  =\left\{
\begin{pmatrix}
b & a\\
a & a+b
\end{pmatrix}
\mid a,b\in\mathbb{Z}\right\}
\]
of the matrix ring $\mathbb{Z}^{2\times2}$.

\begin{enumerate}
\item[\textbf{(a)}] Prove that $A^{2} = A + I_{2}$.

\item[\textbf{(b)}] Prove that the set $\mathcal{F}$ (equipped with the
addition of matrices, the multiplication of matrices, the zero $0_{2 \times2}$
and the unity $I_{2}$) is a commutative ring. \newline(Again, you don't need
to check the ring axioms, as we already know that they hold for arbitrary
matrices and thus all the more for matrices in $\mathcal{F}$. But you do need
to check commutativity of multiplication in $\mathcal{F}$, since it does not
hold for arbitrary matrices. You also need to check that $\mathcal{F}$ is
closed under addition and multiplication and has additive inverses.)
\end{enumerate}

Let $\left(  f_{0},f_{1},f_{2},\ldots\right)  $ be the Fibonacci sequence
(which we have already encountered on
\href{http://www.cip.ifi.lmu.de/~grinberg/t/19s/hw5s.pdf}{homework set \#5}).
Recall that it is defined recursively by
\[
f_{0}=0,\qquad f_{1}=1,\qquad\text{and}\qquad f_{n}=f_{n-1}+f_{n-2}\text{ for
all }n\geq2.
\]

\begin{enumerate}
\item[\textbf{(c)}] Prove that $A^{n} = f_{n} A + f_{n-1} I_{2}$ for all
positive integers $n$.

\item[\textbf{(d)}] Prove that $f_{n+m}=f_{n}f_{m+1}+f_{n-1}f_{m}$ for all
positive integers $n$ and all $m\in\mathbb{N}$.
\end{enumerate}

Now, define a further matrix $B \in\mathcal{F}$ by $B = \left(  -1 \right)  A
+ 1I_{2} = I_{2} - A$.

\begin{enumerate}
\item[\textbf{(e)}] Prove that $B^{2} = B + I_{2}$ and $B^{n} = f_{n} B +
f_{n-1} I_{2}$ for all positive integers $n$.

\item[\textbf{(f)}] Prove that $A^{n} - B^{n} = f_{n} \left(  A - B \right)  $
for all $n \in\mathbb{N}$.

\item[\textbf{(g)}] Prove (again!) that $f_{d}\mid f_{dn}$ for any nonnegative
integers $d$ and $n$.
\end{enumerate}

[\textbf{Hint:} One way to prove \textbf{(d)} is by comparing the $\left(  1,
1 \right)  $-th entries of the two (equal) matrices $A^{n} A^{m+1}$ and
$A^{n+m+1}$, after first using part \textbf{(c)} to expand these matrices.

For part \textbf{(g)}, compare the $\left(  1, 1 \right)  $-th entries of the
matrices $A^{d} - B^{d}$ and $A^{dn} - B^{dn}$, after first proving that
$A^{d} - B^{d} \mid A^{dn} - B^{dn}$ in the commutative ring $\mathcal{F}$.
Note that divisibility is a tricky concept in general rings, but $\mathcal{F}$
is a commutative ring, which lets many arguments from the integer setting go
through unchanged.]

\subsection{Remark}

Contrast the ring $\mathcal{F}$ with the ring $\mathbb{Z}\left[  \phi\right]
$ from Exercise 5 on
\href{http://www.cip.ifi.lmu.de/~grinberg/t/19s/hw5s.pdf}{homework set \#5}.
Both of these rings, as we see, can be used to prove that $f_{d}\mid f_{dn}$
for any nonnegative integers $d$ and $n$. It turns out that these rings have
more in common: they are isomorphic! More precisely, the map%
\begin{align*}
\mathbb{Z}\left[  \phi\right]   &  \rightarrow\mathcal{F},\\
a+b\phi &  \mapsto bA+aI_{2}\qquad\left(  \text{for }a,b\in\mathbb{Z}\right)
\end{align*}
is a ring isomorphism. This makes it less surprising that these rings can
substitute for one another in proving $f_{d}\mid f_{dn}$.

\subsection{Solution sketch}

We first recall that $\mathcal{F}=\left\{  aA+bI_{2}\mid a,b\in\mathbb{Z}%
\right\}  $ (by the definition of $\mathcal{F}$). Hence,%
\begin{equation}
aA+bI_{2}\in\mathcal{F}\qquad\text{for all }a,b\in\mathbb{Z}.
\label{sol.ring.fibonacci.F.aA+binF}%
\end{equation}

\bigskip

\textbf{(a)} This is an easy exercise in multiplying matrices: From $A=%
\begin{pmatrix}
0 & 1\\
1 & 1
\end{pmatrix}
$, we obtain%
\[
A^{2}=%
\begin{pmatrix}
0 & 1\\
1 & 1
\end{pmatrix}
^{2}=%
\begin{pmatrix}
0 & 1\\
1 & 1
\end{pmatrix}%
\begin{pmatrix}
0 & 1\\
1 & 1
\end{pmatrix}
=%
\begin{pmatrix}
1 & 1\\
1 & 2
\end{pmatrix}
.
\]
Comparing this with%
\[
\underbrace{A}_{=%
\begin{pmatrix}
0 & 1\\
1 & 1
\end{pmatrix}
}+\underbrace{I_{2}}_{=%
\begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix}
}=%
\begin{pmatrix}
0 & 1\\
1 & 1
\end{pmatrix}
+%
\begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix}
=%
\begin{pmatrix}
1 & 1\\
1 & 2
\end{pmatrix}
,
\]
we obtain $A^{2}=A+I_{2}$. This solves part \textbf{(a)} of the exercise.

\bigskip

\textbf{(b)} Let us show the following claims:

\begin{statement}
\textit{Claim 1:} We have $0_{2\times2}\in\mathcal{F}$.
\end{statement}

\begin{statement}
\textit{Claim 2:} We have $I_{2}\in\mathcal{F}$.
\end{statement}

\begin{statement}
\textit{Claim 3:} For any $U,V\in\mathcal{F}$, we have $U+V\in\mathcal{F}$ and
$UV\in\mathcal{F}$ and $UV=VU$.
\end{statement}

\begin{statement}
\textit{Claim 4:} For any $U\in\mathcal{F}$, we have $-U\in\mathcal{F}$.
\end{statement}

[\textit{Proof of Claim 1:} We have $\underbrace{0A}_{=0_{2\times2}%
}+\underbrace{0I_{2}}_{=0_{2\times2}}=0_{2\times2}$, so that $0_{2\times
2}=0A+0I_{2}\in\mathcal{F}$ (by \eqref{sol.ring.fibonacci.F.aA+binF}, applied
to $a=0$ and $b=0$). This proves Claim 1.]

[\textit{Proof of Claim 2:} We have $\underbrace{0A}_{=0_{2\times2}%
}+\underbrace{1I_{2}}_{=I_{2}}=I_{2}$, so that $I_{2}=0A+1I_{2}\in\mathcal{F}$
(by \eqref{sol.ring.fibonacci.F.aA+binF}, applied to $a=0$ and $b=1$). This
proves Claim 2.]

[\textit{Proof of Claim 3:} Let $U,V\in\mathcal{F}$. We must prove that
$U+V\in\mathcal{F}$ and $UV\in\mathcal{F}$ and $UV=VU$.

We have $V\in\mathcal{F}=\left\{  aA+bI_{2}\mid a,b\in\mathbb{Z}\right\}
=\left\{  cA+dI_{2}\mid c,d\in\mathbb{Z}\right\}  $ (here, we have renamed the
indices $a$ and $b$ as $c$ and $d$). In other words, there exist some
$c,d\in\mathbb{Z}$ such that $V=cA+dI_{2}$. Consider these $c,d$.

We have $U\in\mathcal{F}=\left\{  aA+bI_{2}\mid a,b\in\mathbb{Z}\right\}  $.
In other words, there exist some $a,b\in\mathbb{Z}$ such that $U=aA+bI_{2}$.
Consider these $a,b$.

From $a,b,c,d\in\mathbb{Z}$, we obtain $a+c\in\mathbb{Z}$ and $b+d\in
\mathbb{Z}$ and $ad+bc+ac\in\mathbb{Z}$ and $ac+bd\in\mathbb{Z}$.

We have $a,b,c,d\in\mathbb{Z}$. Adding the equalities $U=aA+bI_{2}$ and
$V=cA+dI_{2}$ together, we obtain%
\[
U+V=\left(  aA+bI_{2}\right)  +\left(  cA+dI_{2}\right)  =\left(  a+c\right)
A+\left(  b+d\right)  I_{2}\in\mathcal{F}%
\]
(by \eqref{sol.ring.fibonacci.F.aA+binF}, applied to $a+c$ and $b+d$ instead
of $a$ and $b$).

We have $AA=A^{2}=A+I_{2}$ (by part \textbf{(a)} of the exercise).

Multiplying the equalities $U=aA+bI_{2}$ and $V=cA+dI_{2}$ together, we obtain%
\begin{align*}
UV  &  =\left(  aA+bI_{2}\right)  \left(  cA+dI_{2}\right)  =ac\underbrace{AA}%
_{\substack{=A+I_{2}}}+ad\underbrace{AI_{2}}_{=A}+bc\underbrace{I_{2}A}%
_{=A}+bd\underbrace{I_{2}I_{2}}_{=I_{2}}\\
&  =ac\left(  A+I_{2}\right)  +adA+bcA+bdI_{2}=\left(  ad+bc+ac\right)
A+\left(  ac+bd\right)  I_{2}\in\mathcal{F}%
\end{align*}
(by \eqref{sol.ring.fibonacci.F.aA+binF}, applied to $ad+bc+ac$ and $ac+bd$
instead of $a$ and $b$).

Multiplying the equalities $V=cA+dI_{2}$ and $U=aA+bI_{2}$ together, we obtain%
\begin{align*}
VU  &  =\left(  cA+dI_{2}\right)  \left(  aA+bI_{2}\right)  =\underbrace{ca}%
_{=ac}\underbrace{AA}_{\substack{=A+I_{2}}}+\underbrace{cb}_{=bc}%
\underbrace{AI_{2}}_{=A}+\underbrace{da}_{=ad}\underbrace{I_{2}A}%
_{=A}+\underbrace{db}_{=bd}\underbrace{I_{2}I_{2}}_{=I_{2}}\\
&  =ac\left(  A+I_{2}\right)  +bcA+adA+bdI_{2}=\left(  ad+bc+ac\right)
A+\left(  ac+bd\right)  I_{2}.
\end{align*}
Comparing this equality with $UV=\left(  ad+bc+ac\right)  A+\left(
ac+bd\right)  I_{2}$, we obtain $UV=VU$.

Thus, we have proven that $U+V\in\mathcal{F}$ and $UV\in\mathcal{F}$ and
$UV=VU$. This proves Claim 3.]

[\textit{Proof of Claim 4:} Let $U\in\mathcal{F}$. We have $U\in
\mathcal{F}=\left\{  aA+bI_{2}\mid a,b\in\mathbb{Z}\right\}  $. In other
words, there exist some $a,b\in\mathbb{Z}$ such that $U=aA+bI_{2}$. Consider
these $a,b$. Hence,%
\[
-\underbrace{U}_{=aA+bI_{2}}=-\left(  aA+bI_{2}\right)  =\left(  -a\right)
A+\left(  -b\right)  I_{2}\in\mathcal{F}%
\]
(by \eqref{sol.ring.fibonacci.F.aA+binF}, applied to $-a$ and $-b$ instead of
$a$ and $b$). This proves Claim 4.]

Let us now resume solving part \textbf{(b)} of the exercise. Claim 3 shows
that for every $U,V\in\mathcal{F}$, we have $U+V\in\mathcal{F}$. Thus,
addition of matrices defines a binary operation $+$ on $\mathcal{F}$.
Furthermore, Claim 3 shows that for every $U,V\in\mathcal{F}$, we have
$UV\in\mathcal{F}$. Thus, multiplication of matrices defines a binary
operation $\cdot$ on $\mathcal{F}$. Furthermore, $0_{2\times2}\in\mathcal{F}$
(by Claim 1) and $I_{2}\in\mathcal{F}$ (by Claim 2). Hence, we can endow the
set $\mathcal{F}$ with the binary operation $+$ (as addition), the binary
operation $\cdot$ (as multiplication), the element $0_{2\times2}$ (as zero)
and the element $I_{2}$ (as unity). Now, we must prove that the result is a
commutative ring.

Indeed, let us first prove that $\mathcal{F}$ is a ring. To that end, we shall
check all the ring axioms:

\begin{itemize}
\item The \textquotedblleft Existence of additive inverses\textquotedblright%
\ axiom is satisfied, because for every $U\in\mathcal{F}$, there exists an
$U^{\prime}\in\mathcal{F}$ such that $U+U^{\prime}=U^{\prime}+U=0_{2\times2}$.
(Namely, we can set $U^{\prime}=-U$, which is an element of $\mathcal{F}$
because of Claim 4.)

\item All the remaining ring axioms are satisfied, since they are particular
cases of the rules for addition and multiplication of matrices. (For example,
associativity of multiplication holds in $\mathcal{F}$ because it holds for
arbitrary matrices.)
\end{itemize}

\noindent Thus, $\mathcal{F}$ is a ring. Furthermore, every $U,V\in
\mathcal{F}$ satisfy $UV=VU$ (by Claim 3). Hence, the ring $\mathcal{F}$ is
commutative. This solves part \textbf{(b)} of the exercise.

\bigskip

\textbf{(c)} We shall solve part \textbf{(c)} of the exercise by induction on
$n$:

\textit{Induction base:} We have $A^{1}=f_{1}A+f_{1-1}I_{2}$ (since
$\underbrace{f_{1}}_{=1}A+\underbrace{f_{1-1}}_{=f_{0}=0}I_{2}=1A+0I_{2}%
=1A=A=A^{1}$). In other words, part \textbf{(c)} of the exercise holds for
$n=1$. This completes the induction base.

\textit{Induction step:} Let $k$ be a positive integer. Assume that part
\textbf{(c)} of the exercise holds for $n=k$. We must now prove that part
\textbf{(c)} of the exercise holds for $n=k+1$.

We have assumed that part \textbf{(c)} of the exercise holds for $n=k$. In
other words, $A^{k}=f_{k}A+f_{k-1}I_{2}$. But the recursive definition of the
Fibonacci sequence yields $f_{k+1}=f_{k}+f_{k-1}$. Now,%
\begin{align*}
A^{k+1}  &  =A\underbrace{A^{k}}_{=f_{k}A+f_{k-1}I_{2}}=A\left(
f_{k}A+f_{k-1}I_{2}\right)  =f_{k}\underbrace{AA}_{\substack{=A^{2}%
=A+I_{2}\\\text{(by part \textbf{(a)}}\\\text{of the exercise)}}%
}+f_{k-1}\underbrace{AI_{2}}_{=A}=f_{k}\left(  A+I_{2}\right)  +f_{k-1}A\\
&  =\underbrace{\left(  f_{k}+f_{k-1}\right)  }_{=f_{k+1}}A+\underbrace{f_{k}%
}_{=f_{\left(  k+1\right)  -1}}I_{2}=f_{k+1}A+f_{\left(  k+1\right)  -1}I_{2}.
\end{align*}
In other words, part \textbf{(c)} of the exercise holds for $n=k+1$. This
completes the induction step. Hence, part \textbf{(c)} of the exercise is
proven by induction.

\bigskip

\textbf{(d)} Let $n$ be a positive integer. Let $m\in\mathbb{N}$. Part
\textbf{(c)} of the exercise (applied to $n+m+1$ instead of $n$) yields%
\begin{equation}
A^{n+m+1}=f_{n+m+1}A+\underbrace{f_{\left(  n+m+1\right)  -1}}_{=f_{n+m}}%
I_{2}=f_{n+m+1}A+f_{n+m}I_{2}. \label{sol.ring.fibonacci.F.d.1}%
\end{equation}

But part \textbf{(c)} of the exercise (applied to $m+1$ instead of $n$) yields%
\[
A^{m+1}=f_{m+1}A+\underbrace{f_{\left(  m+1\right)  -1}}_{=f_{m}}I_{2}%
=f_{m+1}A+f_{m}I_{2}.
\]
Furthermore, part \textbf{(c)} of the exercise yields%
\[
A^{n}=f_{n}A+f_{n-1}I_{2}.
\]
Multiplying the last two equalities, we obtain%
\begin{align*}
A^{m+1}A^{n}  &  =\left(  f_{m+1}A+f_{m}I_{2}\right)  \left(  f_{n}%
A+f_{n-1}I_{2}\right) \\
&  =\underbrace{f_{m+1}f_{n}}_{=f_{n}f_{m+1}}\underbrace{AA}_{\substack{=A^{2}%
=A+I_{2}\\\text{(by part \textbf{(a)}}\\\text{of the exercise)}}%
}+f_{m+1}f_{n-1}\underbrace{AI_{2}}_{=A}+f_{m}f_{n}\underbrace{I_{2}A}%
_{=A}+\underbrace{f_{m}f_{n-1}}_{=f_{n-1}f_{m}}\underbrace{I_{2}I_{2}}%
_{=I_{2}}\\
&  =f_{n}f_{m+1}\left(  A+I_{2}\right)  +f_{m+1}f_{n-1}A+f_{m}f_{n}%
A+f_{n-1}f_{m}I_{2}\\
&  =\left(  f_{n}f_{m+1}+f_{m+1}f_{n-1}+f_{m}f_{n}\right)  A+\left(
f_{n}f_{m+1}+f_{n-1}f_{m}\right)  I_{2}.
\end{align*}
Comparing this with%
\begin{align*}
A^{m+1}A^{n}  &  =A^{\left(  m+1\right)  +n}\qquad\left(  \text{by the rules
for exponents in the ring }\mathbb{Z}^{2\times2}\right) \\
&  =A^{n+m+1}\qquad\left(  \text{since }\left(  m+1\right)  +n=n+m+1\right) \\
&  =f_{n+m+1}A+f_{n+m}I_{2}\qquad\left(  \text{by
\eqref{sol.ring.fibonacci.F.d.1}}\right)  ,
\end{align*}
we obtain%
\begin{equation}
f_{n+m+1}A+f_{n+m}I_{2}=\left(  f_{n}f_{m+1}+f_{m+1}f_{n-1}+f_{m}f_{n}\right)
A+\left(  f_{n}f_{m+1}+f_{n-1}f_{m}\right)  I_{2}.
\label{sol.ring.fibonacci.F.d.3}%
\end{equation}

We now would certainly want to \textquotedblleft compare the coefficients of
$I_{2}$\textquotedblright\ in this equality, thus concluding that
$f_{n+m}=f_{n}f_{m+1}+f_{n-1}f_{m}$. But why can we do this?

The simplest way to justify this is by comparing the $\left(  1,1\right)  $-th
entries of both matrices. Indeed, for any $u,v\in\mathbb{Z}$, we have%
\[
u\underbrace{A}_{=%
\begin{pmatrix}
0 & 1\\
1 & 1
\end{pmatrix}
}+v\underbrace{I_{2}}_{=%
\begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix}
}=u%
\begin{pmatrix}
0 & 1\\
1 & 1
\end{pmatrix}
+v%
\begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix}
=%
\begin{pmatrix}
v & u\\
u & u+v
\end{pmatrix}
\]
and thus%
\begin{equation}
\left(  \text{the }\left(  1,1\right)  \text{-th entry of the matrix
}uA+vI_{2}\right)  =v. \label{sol.ring.fibonacci.F.d.uv}%
\end{equation}
Applying this equality to $u=f_{n+m+1}$ and $v=f_{n+m}$, we obtain%
\begin{equation}
\left(  \text{the }\left(  1,1\right)  \text{-th entry of the matrix
}f_{n+m+1}A+f_{n+m}I_{2}\right)  =f_{n+m}. \label{sol.ring.fibonacci.F.d.r1}%
\end{equation}
But applying \eqref{sol.ring.fibonacci.F.d.uv} to $u=f_{n}f_{m+1}%
+f_{m+1}f_{n-1}+f_{m}f_{n}$ and $v=f_{n}f_{m+1}+f_{n-1}f_{m}$, we obtain%
\begin{align}
&  \left(  \text{the }\left(  1,1\right)  \text{-th entry of the matrix
}\left(  f_{n}f_{m+1}+f_{m+1}f_{n-1}+f_{m}f_{n}\right)  A+\left(  f_{n}%
f_{m+1}+f_{n-1}f_{m}\right)  I_{2}\right) \nonumber\\
&  =f_{n}f_{m+1}+f_{n-1}f_{m}. \label{sol.ring.fibonacci.F.d.r2}%
\end{align}
But the left hand sides of the equalities \eqref{sol.ring.fibonacci.F.d.r1}
and \eqref{sol.ring.fibonacci.F.d.r2} are equal (because of
\eqref{sol.ring.fibonacci.F.d.3}). Thus, their right hand sides must also be
equal. In other words, we have%
\[
f_{n+m}=f_{n}f_{m+1}+f_{n-1}f_{m}.
\]
This solves part \textbf{(d)} of the exercise.

[\textit{Remark:} There are many other ways to solve this part of the
exercise. For example, if we rename $n$ as $n+1$, then it takes the more
symmetric form $f_{n+m+1}=f_{n+1}f_{m+1}+f_{n}f_{m}$; but this is a well-known
identity (see, e.g., Exercise 3 \textbf{(e)} in
\href{http://www.cip.ifi.lmu.de/~grinberg/t/18s/mt1s.pdf}{UMN Spring 2018 Math
4707 midterm \#1}) and a particular case of \cite[Theorem 2.26 \textbf{(a)}%
]{detnotes}. It can be easily proven by induction on $n$ (or on $m$). Binet's
formula for the Fibonacci numbers also leads to a straightforward solution to
part \textbf{(d)} of the exercise.]

\bigskip

\textbf{(e)} First, we shall show that $B^{2}=B+I_{2}$. Indeed, $B=I_{2}-A$
and thus%
\begin{align*}
B^{2}  &  =\left(  I_{2}-A\right)  ^{2}=\left(  I_{2}-A\right)  \left(
I_{2}-A\right)  =\underbrace{I_{2}\left(  I_{2}-A\right)  }_{=I_{2}%
-A}-\underbrace{A\left(  I_{2}-A\right)  }_{=AI_{2}-AA}\\
&  =I_{2}-A-\left(  AI_{2}-AA\right)  =I_{2}-A-\underbrace{AI_{2}}%
_{=A}+\underbrace{AA}_{\substack{=A^{2}=A+I_{2}\\\text{(by part \textbf{(a)}%
}\\\text{of the exercise)}}}\\
&  =I_{2}-A-A+\left(  A+I_{2}\right)  =2I_{2}-A=\underbrace{\left(
I_{2}-A\right)  }_{=B}+I_{2}=B+I_{2}.
\end{align*}

It remains to prove that $B^{n}=f_{n}B+f_{n-1}I_{2}$ for all positive integers
$n$.

Here is the laziest (but perfectly legitimate) way of doing this: In our
solution to part \textbf{(c)} of this exercise, we have proven that
$A^{n}=f_{n}A+f_{n-1}I_{2}$ for all positive integers $n$. Our proof of this
fact did not use anything specific about the matrix $A$, other than the fact
that $A$ satisfies $A^{2}=A+I_{2}$. Therefore, we can replace each appearance
of \textquotedblleft$A$\textquotedblright\ by \textquotedblleft$B$%
\textquotedblright\ in this proof, and thus obtain a proof of the fact that
$B^{n}=f_{n}B+f_{n-1}I_{2}$ for all positive integers $n$ (because $B$
satisfies $B^{2}=B+I_{2}$). This completes the solution of part \textbf{(e)}
of the exercise.

\bigskip

\textbf{(f)} Let $n\in\mathbb{N}$. We must prove that $A^{n}-B^{n}%
=f_{n}\left(  A-B\right)  $.

If $n=0$, then this is easy\footnote{\textit{Proof.} Assume that $n=0$. Thus,
$A^{n}-B^{n}=\underbrace{A^{0}}_{=I_{2}}-\underbrace{B^{0}}_{=I_{2}}%
=I_{2}-I_{2}=0_{2\times2}$. On the other hand, from $n=0$, we obtain
$f_{n}=f_{0}=0$ and thus $f_{n}\left(  A-B\right)  =0\left(  A-B\right)
=0_{2\times2}$. Comparing this with $A^{n}-B^{n}=0_{2\times2}$, we obtain
$A^{n}-B^{n}=f_{n}\left(  A-B\right)  $. Hence, we have proven that
$A^{n}-B^{n}=f_{n}\left(  A-B\right)  $ under the assumption that $n=0$.}.
Hence, for the rest of this proof, we WLOG assume that $n\neq0$. Thus, $n$ is
a positive integer (since $n\in\mathbb{N}$). Hence, part \textbf{(c)} of this
exercise yields $A^{n}=f_{n}A+f_{n-1}I_{2}$. But part \textbf{(e)} of this
exercise yields $B^{n}=f_{n}B+f_{n-1}I_{2}$. Subtracting the last two
equalities, we obtain%
\[
A^{n}-B^{n}=\left(  f_{n}A+f_{n-1}I_{2}\right)  -\left(  f_{n}B+f_{n-1}%
I_{2}\right)  =f_{n}\left(  A-B\right)  .
\]
This solves part \textbf{(f)} of the exercise.

\bigskip

\textbf{(g)} Given two elements $\alpha$ and $\beta$ of $\mathcal{F}$, we say
that $\alpha\mid\beta$ \textit{in }$\mathcal{F}$ if and only if there exists
some $\gamma\in\mathcal{F}$ such that $\beta=\alpha\gamma$. Thus, we have
defined divisibility in $\mathcal{F}$. Basic properties of divisibility of
integers (such as Proposition 2.2.4 in
\href{http://www.cip.ifi.lmu.de/~grinberg/t/19s/notes.pdf}{the class notes})
still apply to divisibility in $\mathcal{F}$ (with the same proofs), since
$\mathcal{F}$ is a \textbf{commutative} ring.

We recall the following fact (Lemma 2.10.11 \textbf{(a)} in
\href{http://www.cip.ifi.lmu.de/~grinberg/t/19s/notes.pdf}{the class notes}):

\begin{statement}
\textit{Claim 5:} Let $d\in\mathbb{N}$. Let $x$ and $y$ be integers. Then,
$x-y\mid x^{d}-y^{d}$.
\end{statement}

This fact has an analogue for elements of $\mathcal{F}$ instead of integers:

\begin{statement}
\textit{Claim 6:} Let $d\in\mathbb{N}$. Let $x$ and $y$ be elements of
$\mathcal{F}$. Then, $x-y\mid x^{d}-y^{d}$ in $\mathcal{F}$.
\end{statement}

[\textit{Proof of Claim 6:} Both proofs we gave for Claim 5 in the class notes
can be modified in an obvious way to yield proofs of Claim 6, because
$\mathcal{F}$ is a \textbf{commutative} ring.]

Now, let $d$ and $n$ be nonnegative integers. We must prove that $f_{d}\mid
f_{dn}$.

Part \textbf{(f)} of this exercise (applied to $d$ instead of $n$) yields
$A^{d}-B^{d}=f_{d}\left(  A-B\right)  $.

Part \textbf{(f)} of this exercise (applied to $dn$ instead of $n$) yields
$A^{dn}-B^{dn}=f_{dn}\left(  A-B\right)  $.

But $A$ and $B$ are elements of $\mathcal{F}$. Thus, their powers $A^{d}$,
$B^{d}$, $A^{dn}$ and $B^{dn}$ are elements of $\mathcal{F}$ as well (since
$\mathcal{F}$ is a ring). Hence, Claim 2 (applied to $n$, $A^{d}$ and $B^{d}$
instead of $d$, $x$ and $y$) yields $A^{d}-B^{d}\mid\left(  A^{d}\right)
^{n}-\left(  B^{d}\right)  ^{n}$ in $\mathcal{F}$. In view of
\[
A^{d}-B^{d}=f_{d}\left(  A-B\right)  \qquad\text{and}\qquad\left(
A^{d}\right)  ^{n}-\left(  B^{d}\right)  ^{n}=A^{dn}-B^{dn}=f_{dn}\left(
A-B\right)  ,
\]
this rewrites as follows:%
\[
f_{d}\left(  A-B\right)  \mid f_{dn}\left(  A-B\right)  \qquad\text{in
}\mathcal{F}.
\]

Now, it is tempting to \textquotedblleft cancel\textquotedblright\ $A-B$ from
this divisibility, and conclude that $f_{d}\mid f_{dn}$ in $\mathbb{Z}$. To
justify this rigorously, we proceed as follows:

We have $f_{d}\left(  A-B\right)  \mid f_{dn}\left(  A-B\right)  $ in
$\mathcal{F}$. In other words, there exists a matrix $\gamma\in\mathcal{F}$
such that
\begin{equation}
f_{dn}\left(  A-B\right)  =f_{d}\left(  A-B\right)  \gamma
\label{sol.ring.fibonacci.F.g.2}%
\end{equation}
(by the definition of divisibility in $\mathcal{F}$). Consider this $\gamma$.

We have $B=I_{2}-A$ and thus
\[
A-B=A-\left(  I_{2}-A\right)  =2\underbrace{A}_{=%
\begin{pmatrix}
0 & 1\\
1 & 1
\end{pmatrix}
}-\underbrace{I_{2}}_{=%
\begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix}
}=2%
\begin{pmatrix}
0 & 1\\
1 & 1
\end{pmatrix}
-%
\begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix}
=%
\begin{pmatrix}
-1 & 2\\
2 & 1
\end{pmatrix}
.
\]
Hence,%
\begin{equation}
\left(  \text{the }\left(  2,2\right)  \text{-th entry of the matrix
}A-B\right)  =1.\label{sol.ring.fibonacci.F.g.3}%
\end{equation}
Since matrices are scaled entrywise, we now have%
\begin{align}
&  \left(  \text{the }\left(  2,2\right)  \text{-th entry of the matrix
}f_{dn}\left(  A-B\right)  \right)  \nonumber\\
&  =f_{dn}\cdot\underbrace{\left(  \text{the }\left(  2,2\right)  \text{-th
entry of the matrix }A-B\right)  }_{=1}=f_{dn}%
.\label{sol.ring.fibonacci.F.g.5}%
\end{align}
On the other hand, the matrices $A-B$ and $\gamma$ belong to $\mathcal{F}$;
thus, their product $\left(  A-B\right)  \gamma$ belongs to $\mathcal{F}$ as
well, and therefore belongs to $\mathbb{Z}^{2\times2}$ (since $\mathcal{F}%
\subseteq\mathbb{Z}^{2\times2}$). In other words, $\left(  A-B\right)  \gamma$
is a $2\times2$-matrix with integer entries. Hence, each entry of $\left(
A-B\right)  \gamma$ is an integer, i.e., belongs to $\mathbb{Z}$. Thus, in
particular,%
\[
\left(  \text{the }\left(  2,2\right)  \text{-th entry of the matrix }\left(
A-B\right)  \gamma\right)  \in\mathbb{Z}.
\]
Now, \eqref{sol.ring.fibonacci.F.g.5} yields%
\begin{align*}
f_{dn} &  =\left(  \text{the }\left(  2,2\right)  \text{-th entry of the
matrix }\underbrace{f_{dn}\left(  A-B\right)  }_{=f_{d}\left(  A-B\right)
\gamma}\right)  \\
&  =\left(  \text{the }\left(  2,2\right)  \text{-th entry of the matrix
}f_{d}\left(  A-B\right)  \gamma\right)  \\
&  =f_{d}\cdot\left(  \text{the }\left(  2,2\right)  \text{-th entry of the
matrix }\left(  A-B\right)  \gamma\right)
\end{align*}
(since matrices are scaled entrywise). This yields that $f_{d}\mid f_{dn}$ (in
the classical sense of divisibility of integers), because we know that
$\left(  \text{the }\left(  2,2\right)  \text{-th entry of the matrix }\left(
A-B\right)  \gamma\right)  \in\mathbb{Z}$. This solves part \textbf{(g)} of
the exercise.

[\textit{Remark:} Once again, there are other ways to solve this part of the
exercise. For example, it can be restated as \textquotedblleft Prove that
$f_{u}\mid f_{v}$ whenever $u$ and $v$ are nonnegative integers satisfying
$u\mid v$\textquotedblright; but in this form, it is clearly a particular case
of \cite[Theorem 2.26 \textbf{(c)}]{detnotes}.]

%----------------------------------------------------------------------------------------
%	EXERCISE 6
%----------------------------------------------------------------------------------------
\rule{\linewidth}{0.3pt} \\[0.4cm]

\section{Exercise 6: ISBNs vs. fat fingers}

\subsection{Problem}

An \textit{ISBN} shall mean a $10$-tuple $\left(  a_{1},a_{2},\ldots
,a_{10}\right)  \in\left\{  0,1,\ldots,10\right\}  ^{10}$ such that
\[
1a_{1}+2a_{2}+\cdots+10a_{10}\equiv0\operatorname{mod}11.
\]

\noindent(For example, the $10$-tuple $\left(  1, 1, \ldots, 1 \right)  $ is
an ISBN.)

Prove the following:

\begin{enumerate}
\item[\textbf{(a)}] If $\mathbf{a} = \left(  a_{1}, a_{2}, \ldots, a_{10}
\right)  $ and $\mathbf{b} = \left(  b_{1}, b_{2}, \ldots, b_{10} \right)  $
are two ISBNs that are equal in all but one entry (i.e., there exists some $k
\in\left\{  1, 2, \ldots, 10 \right\}  $ such that $a_{i} = b_{i}$ for all $i
\neq k$), then $\mathbf{a} = \mathbf{b}$.

\item[\textbf{(b)}] If an ISBN $\mathbf{a} = \left(  a_{1}, a_{2}, \ldots,
a_{10} \right)  $ is obtained from an ISBN $\mathbf{b} = \left(  b_{1}, b_{2},
\ldots, b_{10} \right)  $ by swapping two entries (i.e., there exist $k,
\ell\in\left\{  1, 2, \ldots, 10 \right\}  $ such that $a_{k} = b_{\ell}$ and
$a_{\ell}= b_{k}$ and $a_{i} = b_{i}$ for all $i \notin\left\{  k,
\ell\right\}  $), then $\mathbf{a} = \mathbf{b}$.
\end{enumerate}

\subsection{Remark}

What we called ISBN here is essentially the definition of an
\href{https://en.wikipedia.org/wiki/International_Standard_Book_Number#ISBN-10_check_digits}{ISBN-10}
-- an international standard for book identifiers used from the 1970s until
2007. For example, the ISBN-10 of the Graham/Knuth/Patashnik book ``Concrete
Mathematics'' is ``0-201-55802-5'', which corresponds to $\left(  0, 2, 0, 1,
5, 5, 8, 0, 2, 5 \right)  $; you can check that this is indeed an ISBN
according to our definition.

(An ``X'' in a real-life ISBN stands for an entry that is $10$.)

As this exercise shows, ISBNs have an error-detection property: If you make a
typo in a single digit or accidentally swap two digits, the result will not be
an ISBN, so you will know that something has gone wrong. This helps you avoid
ordering the wrong book from a bookstore or library.
\href{https://en.wikipedia.org/wiki/Luhn_algorithm}{Credit card numbers have a
similar error-detection feature}.

This is one of the simplest examples of an
\href{https://en.wikipedia.org/wiki/Error_correction_code}{error correction
code}. We may or may not see more of them in class. For now, you can think
about how to define ``ISBNs''

\begin{itemize}
\item in $\left\{  0, 1, \ldots, 4 \right\}  ^{4}$;

\item in $\left\{  0, 1, \ldots, 6 \right\}  ^{6}$;

\item in $\left\{  0, 1, \ldots, 8 \right\}  ^{8}$ (this is harder!).
\end{itemize}

\subsection{Solution sketch}

Before we solve the exercise, let us prove a simple claim which our solution
will rest upon:

\begin{statement}
\textit{Claim 1:} Let $u$ and $v$ be two elements of $\left\{  -10,-9,\ldots
,10\right\}  $ such that $uv\equiv0\operatorname{mod}11$ and $u\neq0$. Then,
$v=0$.
\end{statement}

[\textit{Proof of Claim 1:} Assume the contrary. Thus, $v\neq0$.

We have $u\in\left\{  -10,-9,\ldots,10\right\}  $, thus $-10\leq u\leq10$ and
therefore $\left\vert u\right\vert \leq10$. In other words, $10\geq\left\vert
u\right\vert $.

If we had $11\mid u$, then Proposition 2.2.3 \textbf{(b)} in
\href{http://www.cip.ifi.lmu.de/~grinberg/t/19s/notes.pdf}{the class notes}
(applied to $a=11$ and $b=u$) would yield $\left\vert 11\right\vert
\leq\left\vert u\right\vert $ (since $u\neq0$); but this would contradict
$\left\vert 11\right\vert =11>10\geq\left\vert u\right\vert $. Hence, we
cannot have $11\mid u$. The same argument (applied to $v$ instead of $u$)
shows that we cannot have $11\mid v$ (since $v\neq0$). Thus, neither $11\mid
u$ nor $11\mid v$ holds.

But $11$ is a prime, and we have $11\mid uv$ (since $uv\equiv
0\operatorname{mod}11$). Thus, Theorem 2.13.6 in
\href{http://www.cip.ifi.lmu.de/~grinberg/t/19s/notes.pdf}{the class notes}
(applied to $p=11$, $a=u$ and $b=v$) shows that $11\mid u$ or $11\mid v$. This
contradicts the fact that neither $11\mid u$ nor $11\mid v$ holds. This
contradiction shows that our assumption was false. Hence, Claim 1 is proven.]

\bigskip

\textbf{(a)} Let $\mathbf{a}=\left(  a_{1},a_{2},\ldots,a_{10}\right)  $ and
$\mathbf{b}=\left(  b_{1},b_{2},\ldots,b_{10}\right)  $ be two ISBNs that are
equal in all but one entry. We must prove that $\mathbf{a}=\mathbf{b}$.

We have assumed that $\mathbf{a}$ and $\mathbf{b}$ are equal in all but one
entry. In other words, there exists some $k\in\left\{  1,2,\ldots,10\right\}
$ such that
\begin{equation}
a_{i}=b_{i}\qquad\text{for all }i\neq k. \label{sol.ent.isbn.a.ass}%
\end{equation}
Consider this $k$.

We have assumed that $\mathbf{a}$ is an ISBN. In other words, $\left(
a_{1},a_{2},\ldots,a_{10}\right)  \in\left\{  0,1,\ldots,10\right\}  ^{10}$
and $1a_{1}+2a_{2}+\cdots+10a_{10}\equiv0\operatorname{mod}11$.

Thus,%
\begin{equation}
\sum_{i\in\left\{  1,2,\ldots,10\right\}  }ia_{i}=1a_{1}+2a_{2}+\cdots
+10a_{10}\equiv0\operatorname{mod}11. \label{sol.ent.isbn.a.sum-iai}%
\end{equation}
Similarly,%
\begin{equation}
\sum_{i\in\left\{  1,2,\ldots,10\right\}  }ib_{i}\equiv0\operatorname{mod}11.
\label{sol.ent.isbn.a.sum-ibi}%
\end{equation}
But%
\begin{align*}
\sum_{i\in\left\{  1,2,\ldots,10\right\}  }ia_{i}  &  =ka_{k}+\sum
_{\substack{i\in\left\{  1,2,\ldots,10\right\}  ;\\i\neq k}}i\underbrace{a_{i}%
}_{\substack{=b_{i}\\\text{(by \eqref{sol.ent.isbn.a.ass})}}}\\
&  \qquad\left(  \text{here, we have split off the addend for }i=k\text{ from
the sum}\right) \\
&  =ka_{k}+\sum_{\substack{i\in\left\{  1,2,\ldots,10\right\}  ;\\i\neq
k}}ib_{i},
\end{align*}
so that \eqref{sol.ent.isbn.a.sum-iai} rewrites as%
\begin{equation}
ka_{k}+\sum_{\substack{i\in\left\{  1,2,\ldots,10\right\}  ;\\i\neq k}%
}ib_{i}\equiv0\operatorname{mod}11. \label{sol.ent.isbn.a.sum-iai2}%
\end{equation}
Furthermore,%
\begin{align*}
\sum_{i\in\left\{  1,2,\ldots,10\right\}  }ib_{i}  &  =kb_{k}+\sum
_{\substack{i\in\left\{  1,2,\ldots,10\right\}  ;\\i\neq k}}ib_{i}\\
&  \qquad\left(  \text{here, we have split off the addend for }i=k\text{ from
the sum}\right)  ,
\end{align*}
so that \eqref{sol.ent.isbn.a.sum-ibi} rewrites as%
\[
kb_{k}+\sum_{\substack{i\in\left\{  1,2,\ldots,10\right\}  ;\\i\neq k}%
}ib_{i}\equiv0\operatorname{mod}11.
\]
Subtracting this congruence from the congruence
\eqref{sol.ent.isbn.a.sum-iai2}, we obtain
\[
\left(  ka_{k}+\sum_{\substack{i\in\left\{  1,2,\ldots,10\right\}  ;\\i\neq
k}}ib_{i}\right)  -\left(  kb_{k}+\sum_{\substack{i\in\left\{  1,2,\ldots
,10\right\}  ;\\i\neq k}}ib_{i}\right)  \equiv0-0=0\operatorname{mod}11.
\]
In view of
\[
\left(  ka_{k}+\sum_{\substack{i\in\left\{  1,2,\ldots,10\right\}  ;\\i\neq
k}}ib_{i}\right)  -\left(  kb_{k}+\sum_{\substack{i\in\left\{  1,2,\ldots
,10\right\}  ;\\i\neq k}}ib_{i}\right)  =ka_{k}-kb_{k}=k\left(  a_{k}%
-b_{k}\right)  ,
\]
this rewrites as $k\left(  a_{k}-b_{k}\right)  \equiv0\operatorname{mod}11$.

We have $a_{k}\in\left\{  0,1,\ldots,10\right\}  $ (since $\left(  a_{1}%
,a_{2},\ldots,a_{10}\right)  \in\left\{  0,1,\ldots,10\right\}  ^{10}$) and
$b_{k}\in\left\{  0,1,\ldots,10\right\}  $ (similarly). Hence, $a_{k}-b_{k}%
\in\left\{  -10,-9,\ldots,10\right\}  $. Furthermore, $k\in\left\{
1,2,\ldots,10\right\}  \subseteq\left\{  -10,-9,\ldots,10\right\}  $ and
$k\neq0$. Thus, Claim 1 (applied to $u=k$ and $v=a_{k}-b_{k}$) yields
$a_{k}-b_{k}=0$. In other words, $a_{k}=b_{k}$.

Now, \eqref{sol.ent.isbn.a.ass} shows that any entry of the $10$-tuple
$\mathbf{a}$ is equal to the corresponding entry of the $10$-tuple
$\mathbf{b}$, except perhaps the $k$-th entry. But the equality $a_{k}=b_{k}$
shows that the $k$-th entries of these two $10$-tuples $\mathbf{a}$ and
$\mathbf{b}$ are also equal to each other. Thus, each entry of $\mathbf{a}$ is
equal to the corresponding entry of $\mathbf{b}$. In other words,
$\mathbf{a}=\mathbf{b}$. This solves part \textbf{(a)} of the exercise.

\bigskip

\textbf{(b)} Let $\mathbf{a}=\left(  a_{1},a_{2},\ldots,a_{10}\right)  $ and
$\mathbf{b}=\left(  b_{1},b_{2},\ldots,b_{10}\right)  $ be two ISBNs such that
$\mathbf{a}$ is obtained from $\mathbf{b}$ by swapping two entries. We must
prove that $\mathbf{a}=\mathbf{b}$.

We have assumed that $\mathbf{a}$ is obtained from $\mathbf{b}$ by swapping
two entries. In other words, there exist $k,\ell\in\left\{  1,2,\ldots
,10\right\}  $ such that $a_{k}=b_{\ell}$ and $a_{\ell}=b_{k}$ and
\begin{equation}
a_{i}=b_{i}\qquad\text{for all }i\notin\left\{  k,\ell\right\}  .
\label{sol.ent.isbn.b.ass}%
\end{equation}
Consider these $k,\ell$.

We must prove that $\mathbf{a}=\mathbf{b}$. If $a_{k}=a_{\ell}$, then this is
true\footnote{\textit{Proof.} Assume that $a_{k}=a_{\ell}$. Thus,
$a_{k}=a_{\ell}=b_{k}$ and $a_{\ell}=a_{k}=b_{\ell}$. Now, from
\eqref{sol.ent.isbn.b.ass}, we know that the equality $a_{i}=b_{i}$ holds for
all $i\notin\left\{  k,\ell\right\}  $. But this equality also holds for $i=k$
(since $a_{k}=b_{k}$) and for $i=\ell$ (since $a_{\ell}=b_{\ell}$). Thus, this
equality holds for all $i\in\left\{  1,2,\ldots,10\right\}  $. In other words,
$\mathbf{a}=\mathbf{b}$, qed.}. Hence, for the rest of this solution, we WLOG
assume that $a_{k}\neq a_{\ell}$. Thus, $k\neq\ell$, so that $k-\ell\neq0$.

Also, we have $a_{k}\neq a_{\ell}$. In view of $a_{k}=b_{\ell}$ and $a_{\ell
}=b_{k}$, this rewrites as $b_{\ell}\neq b_{k}$.

We have assumed that $\mathbf{a}$ is an ISBN. In other words, $\left(
a_{1},a_{2},\ldots,a_{10}\right)  \in\left\{  0,1,\ldots,10\right\}  ^{10}$
and $1a_{1}+2a_{2}+\cdots+10a_{10}\equiv0\operatorname{mod}11$.

Thus,%
\begin{equation}
\sum_{i\in\left\{  1,2,\ldots,10\right\}  }ia_{i}=1a_{1}+2a_{2}+\cdots
+10a_{10}\equiv0\operatorname{mod}11. \label{sol.ent.isbn.b.sum-iai}%
\end{equation}
Similarly,%
\begin{equation}
\sum_{i\in\left\{  1,2,\ldots,10\right\}  }ib_{i}\equiv0\operatorname{mod}11.
\label{sol.ent.isbn.b.sum-ibi}%
\end{equation}
But%
\begin{align*}
\sum_{i\in\left\{  1,2,\ldots,10\right\}  }ia_{i}  &  =k\underbrace{a_{k}%
}_{=b_{\ell}}+\ell\underbrace{a_{\ell}}_{=b_{k}}+\sum_{\substack{i\in\left\{
1,2,\ldots,10\right\}  ;\\i\notin\left\{  k,\ell\right\}  }}i\underbrace{a_{i}%
}_{\substack{=b_{i}\\\text{(by \eqref{sol.ent.isbn.b.ass})}}}\\
&  \qquad\left(
\begin{array}
[c]{c}%
\text{here, we have split off the addends for }i=k\text{ and for }i=\ell\text{
from the}\\
\text{sum (and these were indeed two different addends, since }k\neq
\ell\text{)}%
\end{array}
\right) \\
&  =kb_{\ell}+\ell b_{k}+\sum_{\substack{i\in\left\{  1,2,\ldots,10\right\}
;\\i\notin\left\{  k,\ell\right\}  }}ib_{i},
\end{align*}
so that \eqref{sol.ent.isbn.b.sum-iai} rewrites as%
\begin{equation}
kb_{\ell}+\ell b_{k}+\sum_{\substack{i\in\left\{  1,2,\ldots,10\right\}
;\\i\notin\left\{  k,\ell\right\}  }}ib_{i}\equiv0\operatorname{mod}11.
\label{sol.ent.isbn.b.sum-iai2}%
\end{equation}
Furthermore,%
\begin{align*}
\sum_{i\in\left\{  1,2,\ldots,10\right\}  }ia_{i}  &  =kb_{k}+\ell b_{\ell
}+\sum_{\substack{i\in\left\{  1,2,\ldots,10\right\}  ;\\i\notin\left\{
k,\ell\right\}  }}ib_{i}\\
&  \qquad\left(
\begin{array}
[c]{c}%
\text{here, we have split off the addends for }i=k\text{ and for }i=\ell\text{
from the}\\
\text{sum (and these were indeed two different addends, since }k\neq
\ell\text{)}%
\end{array}
\right)  ,
\end{align*}
so that \eqref{sol.ent.isbn.b.sum-ibi} rewrites as%
\[
kb_{k}+\ell b_{\ell}+\sum_{\substack{i\in\left\{  1,2,\ldots,10\right\}
;\\i\notin\left\{  k,\ell\right\}  }}ib_{i}\equiv0\operatorname{mod}11.
\]
Subtracting this congruence from the congruence
\eqref{sol.ent.isbn.b.sum-iai2}, we obtain
\[
\left(  kb_{\ell}+\ell b_{k}+\sum_{\substack{i\in\left\{  1,2,\ldots
,10\right\}  ;\\i\notin\left\{  k,\ell\right\}  }}ib_{i}\right)  -\left(
kb_{k}+\ell b_{\ell}+\sum_{\substack{i\in\left\{  1,2,\ldots,10\right\}
;\\i\notin\left\{  k,\ell\right\}  }}ib_{i}\right)  \equiv
0-0=0\operatorname{mod}11.
\]
In view of
\begin{align*}
\left(  kb_{\ell}+\ell b_{k}+\sum_{\substack{i\in\left\{  1,2,\ldots
,10\right\}  ;\\i\notin\left\{  k,\ell\right\}  }}ib_{i}\right)  -\left(
kb_{k}+\ell b_{\ell}+\sum_{\substack{i\in\left\{  1,2,\ldots,10\right\}
;\\i\notin\left\{  k,\ell\right\}  }}ib_{i}\right)   &  =kb_{\ell}+\ell
b_{k}-kb_{k}-\ell b_{\ell}\\
&  =\left(  k-\ell\right)  \left(  b_{\ell}-b_{k}\right)  ,
\end{align*}
this rewrites as $\left(  k-\ell\right)  \left(  b_{\ell}-b_{k}\right)
\equiv0\operatorname{mod}11$.

But $\mathbf{b}$ is an ISBN; thus, $\left(  b_{1},b_{2},\ldots,b_{10}\right)
\in\left\{  0,1,\ldots,10\right\}  ^{10}$. Hence, $b_{k}\in\left\{
0,1,\ldots,10\right\}  $ and $b_{\ell}\in\left\{  0,1,\ldots,10\right\}  $.
Therefore, $b_{\ell}-b_{k}\in\left\{  -10,-9,\ldots,10\right\}  $.
Furthermore, $k-\ell\in\left\{  -10,-9,\ldots,10\right\}  $ (since both $k$
and $\ell$ belong to the set $\left\{  1,2,\ldots,10\right\}  $) and
$k-\ell\neq0$ (since $k\neq\ell$). Thus, Claim 1 (applied to $u=k-\ell$ and
$v=b_{\ell}-b_{k}$) yields $b_{\ell}-b_{k}=0$. In other words, $b_{\ell}%
=b_{k}$. This contradicts $b_{\ell}\neq b_{k}$. Thus, $\mathbf{a}=\mathbf{b}$
(because \textit{ex falso quodlibet})\footnote{Of course, this shows that the
case we are considering doesn't ever happen -- i.e., our WLOG assumption left
us only an impossible case to consider. But it was not clear a priori that
this case was impossible; we had to work to reach this conclusion.}. This
solves part \textbf{(b)} of the exercise.

\begin{thebibliography}{99999999}                                                                                %

\bibitem[Grinbe19]{detnotes}Darij Grinberg, \textit{Notes on the combinatorial
fundamentals of algebra}, 10 January 2019. \newline%
\url{http://www.cip.ifi.lmu.de/~grinberg/primes2015/sols.pdf} \newline The
numbering of theorems and formulas in this link might shift when the project
gets updated; for a \textquotedblleft frozen\textquotedblright\ version whose
numbering is guaranteed to match that in the citations above, see
\url{https://github.com/darijgr/detnotes/releases/tag/2019-01-10} .

\bibitem[Rogers71]{Rogers71}\href{https://doi.org/10.1007/BF02993626}{Kenneth
Rogers, \textit{An elementary proof of a theorem of Jacobson}, Abhandlungen
Aus Dem Mathematischen Seminar Der Universit\"{a}t Hamburg, \textbf{35} (3-4),
1971, pp. 223--229. doi:10.1007/bf02993626}.

\bibitem[Vorobi02]{Vorobi02}%
\href{https://doi.org/10.1007/978-3-0348-8107-4}{Nicolai N. Vorobiev,
\textit{Fibonacci Numbers}, Translated from the Russian by Mircea Martin,
Springer 2002 (translation of the 6th Russian edition)}.
\end{thebibliography}

\end{document}