# Tagged: Cayley-Hamilton theorem

## Problem 537

Let $A$ and $B$ be $2\times 2$ matrices such that $(AB)^2=O$, where $O$ is the $2\times 2$ zero matrix.

Determine whether $(BA)^2$ must be $O$ as well. If so, prove it. If not, give a counter example.

## Problem 505

Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix.
Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula:
$(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.$

Using the formula, calculate the inverse matrix of $\begin{bmatrix} 2 & 1\\ 1& 2 \end{bmatrix}$.

## Problem 502

Find the inverse matrix of the $3\times 3$ matrix
$A=\begin{bmatrix} 7 & 2 & -2 \\ -6 &-1 &2 \\ 6 & 2 & -1 \end{bmatrix}$ using the Cayley-Hamilton theorem.

## Problem 471

Let $A$ be a $3\times 3$ real orthogonal matrix with $\det(A)=1$.

(a) If $\frac{-1+\sqrt{3}i}{2}$ is one of the eigenvalues of $A$, then find the all the eigenvalues of $A$.

(b) Let
$A^{100}=aA^2+bA+cI,$ where $I$ is the $3\times 3$ identity matrix.
Using the Cayley-Hamilton theorem, determine $a, b, c$.

(Kyushu University, Linear Algebra Exam Problem)

## Problem 421

Find the inverse matrix of the matrix
$A=\begin{bmatrix} 1 & 1 & 2 \\ 9 &2 &0 \\ 5 & 0 & 3 \end{bmatrix}$ using the Cayley–Hamilton theorem.

## Problem 375

Let $n>1$ be a positive integer. Let $V=M_{n\times n}(\C)$ be the vector space over the complex numbers $\C$ consisting of all complex $n\times n$ matrices. The dimension of $V$ is $n^2$.
Let $A \in V$ and consider the set
$S_A=\{I=A^0, A, A^2, \dots, A^{n^2-1}\}$ of $n^2$ elements.
Prove that the set $S_A$ cannot be a basis of the vector space $V$ for any $A\in V$.

## Problem 348

Let $A$ be an $n\times n$ complex matrix.
Let $p(x)=\det(xI-A)$ be the characteristic polynomial of $A$ and write it as
$p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,$ where $a_i$ are real numbers.

Let $C$ be the companion matrix of the polynomial $p(x)$ given by
$C=\begin{bmatrix} 0 & 0 & \dots & 0 &-a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & & \ddots & & \vdots \\ 0 & 0 & \dots & 1 & -a_{n-1} \end{bmatrix}= [\mathbf{e}_2, \mathbf{e}_3, \dots, \mathbf{e}_n, -\mathbf{a}],$ where $\mathbf{e}_i$ is the unit vector in $\C^n$ whose $i$-th entry is $1$ and zero elsewhere, and the vector $\mathbf{a}$ is defined by
$\mathbf{a}=\begin{bmatrix} a_0 \\ a_1 \\ \vdots \\ a_{n-1} \end{bmatrix}.$

Then prove that the following two statements are equivalent.

1. There exists a vector $\mathbf{v}\in \C^n$ such that
$\mathbf{v}, A\mathbf{v}, A^2\mathbf{v}, \dots, A^{n-1}\mathbf{v}$ form a basis of $\C^n$.
2. There exists an invertible matrix $S$ such that $S^{-1}AS=C$.
(Namely, $A$ is similar to the companion matrix of its characteristic polynomial.)

## Problem 337

Let $A, B$ be complex $2\times 2$ matrices satisfying the relation
$A=AB-BA.$

Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix.

## Problem 323

Suppose that $A$ is $2\times 2$ matrix that has eigenvalues $-1$ and $3$.
Then for each positive integer $n$ find $a_n$ and $b_n$ such that
$A^{n+1}=a_nA+b_nI,$ where $I$ is the $2\times 2$ identity matrix.

## Problem 191

Let
$A=\begin{bmatrix} 1 & -1\\ 2& 3 \end{bmatrix}.$

Find the eigenvalues and the eigenvectors of the matrix
$B=A^4-3A^3+3A^2-2A+8E.$

(Nagoya University Linear Algebra Exam Problem)

## 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 11

An $n\times n$ matrix $A$ is called nilpotent if $A^k=O$, where $O$ is the $n\times n$ zero matrix.
Prove the followings.

(a) The matrix $A$ is nilpotent if and only if all the eigenvalues of $A$ is zero.

(b) The matrix $A$ is nilpotent if and only if $A^n=O$.