## Problem 58

Let $A$ be an $n \times n$ matrix over a field $K$. Prove that
$\rk(A^2)-\rk(A^3)\leq \rk(A)-\rk(A^2),$ where $\rk(B)$ denotes the rank of a matrix $B$.

(University of California, Berkeley, Qualifying Exam)

## Problem 57

Let $A$ and $B$ be $n \times n$ matrices.

Prove that the characteristic polynomials for the matrices $AB$ and $BA$ are the same.

## Problem 56

Suppose that $A$ is an $n\times n$ singular matrix.
Prove that for sufficiently small $\epsilon>0$, the matrix $A-\epsilon I$ is nonsingular, where $I$ is the $n \times n$ identity matrix.

## Problem 55

Let $A$ and $B$ are $n \times n$ matrices with real entries.
Assume that $A+B$ is invertible. Then show that
$A(A+B)^{-1}B=B(A+B)^{-1}A.$

(University of California, Berkeley Qualifying Exam)

## Problem 54

Determine all the conjugacy classes of the dihedral group
$D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle$ of order $8$.

## Problem 53

Let $D_8$ be the dihedral group of order $8$.
Using the generators and relations, we have
$D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle.$

(a) Let $A$ be the subgroup of $D_8$ generated by $r$, that is, $A=\{1,r,r^2,r^3\}$.
Prove that the centralizer $C_{D_8}(A)=A$.

(b) Show that the normalizer $N_{D_8}(A)=D_8$.

(c) Show that the center $Z(D_8)=\langle r^2 \rangle=\{1,r^2\}$, the subgroup generated by $r^2$.

## Problem 52

Let $n$ be a positive integer. Let $D_{2n}$ be the dihedral group of order $2n$. Using the generators and the relations, the dihedral group $D_{2n}$ is given by
$D_{2n}=\langle r,s \mid r^n=s^2=1, sr=r^{-1}s\rangle.$ Put $\theta=2 \pi/n$.

(a) Prove that the matrix $\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix}$ is the matrix representation of the linear transformation $T$ which rotates the $x$-$y$ plane about the origin in a counterclockwise direction by $\theta$ radians.

(b) Let $\GL_2(\R)$ be the group of all $2 \times 2$ invertible matrices with real entries. Show that the map $\rho: D_{2n} \to \GL_2(\R)$ defined on the generators by
$\rho(r)=\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix} \text{ and } \rho(s)=\begin{bmatrix} 0 & 1\\ 1& 0 \end{bmatrix}$ extends to a homomorphism of $D_{2n}$ into $\GL_2(\R)$.

(c) Determine whether the homomorphism $\rho$ in part (b) is injective and/or surjective.

## Problem 51

Let $A$ and $B$ be an $n \times n$ matrices.
Suppose that all the eigenvalues of $A$ are distinct and the matrices $A$ and $B$ commute, that is $AB=BA$.

Then prove that each eigenvector of $A$ is an eigenvector of $B$.

(It could be that each eigenvector is an eigenvector for distinct eigenvalues.)

## Problem 50

Let
$A=\begin{bmatrix} \frac{1}{7} & \frac{3}{7} & \frac{3}{7} \\ \frac{3}{7} &\frac{1}{7} &\frac{3}{7} \\ \frac{3}{7} & \frac{3}{7} & \frac{1}{7} \end{bmatrix}$ be $3 \times 3$ matrix. Find

$\lim_{n \to \infty} A^n.$

(Nagoya University Linear Algebra Exam)

## Problem 49

Let $A$ and $B$ be normal subgroups of a group $G$. Suppose $A\cap B=\{e\}$, where $e$ is the unit element of the group $G$.
Show that for any $a \in A$ and $b \in B$ we have $ab=ba$.

## Problem 48

Let $V$ be an $n$-dimensional vector space over a field $K$.
Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ are linearly independent vectors in $V$.

Are the following vectors linearly independent?

$\mathbf{v}_1+\mathbf{v}_2, \quad \mathbf{v}_2+\mathbf{v}_3, \quad \dots, \quad \mathbf{v}_{k-1}+\mathbf{v}_k, \quad \mathbf{v}_k+\mathbf{v}_1.$

If it is linearly dependent, give a non-trivial linear combination of these vectors summing up to the zero vector.

## Problem 47

Let $T=\begin{bmatrix} 1 & 0 & 2 \\ 0 &1 &1 \\ 0 & 0 & 2 \end{bmatrix}$.
Calculate and simplify the expression
$-T^3+4T^2+5T-2I,$ where $I$ is the $3\times 3$ identity matrix.

(The Ohio State University Linear Algebra Exam)

## Problem 46

Let $A$ be an $n\times n$ matrix such that $A^k=I_n$, where $k\in \N$ and $I_n$ is the $n \times n$ identity matrix.

Show that the trace of $(A^{-1})^{\trans}$ is the conjugate of the trace of $A$. That is, show that $\tr((A^{-1})^{\trans})=\overline{\tr(A)}$.

## Problem 45

Calculate the determinants of the following $n\times n$ matrices.
$A=\begin{bmatrix} 1 & 0 & 0 & \dots & 0 & 0 &1 \\ 1 & 1 & 0 & \dots & 0 & 0 & 0 \\ 0 & 1 & 1 & \dots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \dots & \dots & \ddots & \vdots \\ 0 & 0 & 0 &\dots & 1 & 1 & 0\\ 0 & 0 & 0 &\dots & 0 & 1 & 1 \end{bmatrix}$

The entries of $A$ is $1$ at the diagonal entries, entries below the diagonal, and $(1, n)$-entry.
The other entries are zero.
$B=\begin{bmatrix} 1 & 0 & 0 & \dots & 0 & 0 & -1 \\ -1 & 1 & 0 & \dots & 0 & 0 & 0 \\ 0 & -1 & 1 & \dots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \dots & \dots & \ddots & \vdots \\ 0 & 0 & 0 &\dots & -1 & 1 & 0\\ 0 & 0 & 0 &\dots & 0 & -1 & 1 \end{bmatrix}.$

The entries of $B$ is $1$ at the diagonal entries.
The entries below the diagonal and $(1,n)$-entry are $-1$.
The other entries are zero.

## Problem 44

Suppose that a real matrix $A$ maps each of the following vectors
$\mathbf{x}_1=\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \mathbf{x}_2=\begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \mathbf{x}_3=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$ into the vectors
$\mathbf{y}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \mathbf{y}_2=\begin{bmatrix} -1 \\ 0 \\ 3 \end{bmatrix}, \mathbf{y}_3=\begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix},$ respectively.
That is, $A\mathbf{x}_i=\mathbf{y}_i$ for $i=1,2,3$.
Find the matrix $A$.

(Kyoto University Exam)

## Problem 43

Let $a$ and $b$ be two distinct positive real numbers. Define matrices
$A:=\begin{bmatrix} 0 & a\\ a & 0 \end{bmatrix}, \,\, B:=\begin{bmatrix} 0 & b\\ b& 0 \end{bmatrix}.$

Find all the pairs $(\lambda, X)$, where $\lambda$ is a real number and $X$ is a non-zero real matrix satisfying the relation
$AX+XB=\lambda X. \tag{*}$

(The University of Tokyo Linear Algebra Exam)

## Problem 42

Let $A$ be a $4\times 4$ real symmetric matrix. Suppose that $\mathbf{v}_1=\begin{bmatrix} -1 \\ 2 \\ 0 \\ -1 \end{bmatrix}$ is an eigenvector corresponding to the eigenvalue $1$ of $A$.
Suppose that the eigenspace for the eigenvalue $2$ is $3$-dimensional.

(a) Find an orthonormal basis for the eigenspace of the eigenvalue $2$ of $A$.

(b) Find $A\mathbf{v}$, where
$\mathbf{v}=\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}.$

(The University of Tokyo Linear Algebra Exam)

## Problem 41

Find $A^{10}$, where $A=\begin{bmatrix} 4 & 3 & 0 & 0 \\ 3 &-4 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{bmatrix}$.

(Harvard University Exam)

## Problem 40

Find a basis for the subspace $W$ of all vectors in $\R^4$ which are perpendicular to the columns of the matrix
$A=\begin{bmatrix} 11 & 12 & 13 & 14 \\ 21 &22 & 23 & 24 \\ 31 & 32 & 33 & 34 \\ 41 & 42 & 43 & 44 \end{bmatrix}.$

(Harvard University Exam)

## Problem 39

Suppose that $A$ is a diagonalizable matrix with characteristic polynomial
$f_A(\lambda)=\lambda^2(\lambda-3)(\lambda+2)^3(\lambda-4)^3.$

(a) Find the size of the matrix $A$.

(b) Find the dimension of $E_4$, the eigenspace corresponding to the eigenvalue $\lambda=4$.

(c) Find the dimension of the kernel(nullspace) of $A$.

(Stanford University Linear Algebra Exam)