## An Example of a Matrix that Cannot Be a Commutator

## Problem 565

Let $I$ be the $2\times 2$ identity matrix.

Then prove that $-I$ cannot be a commutator $[A, B]:=ABA^{-1}B^{-1}$ for any $2\times 2$ matrices $A$ and $B$ with determinant $1$.

of the day

Let $I$ be the $2\times 2$ identity matrix.

Then prove that $-I$ cannot be a commutator $[A, B]:=ABA^{-1}B^{-1}$ for any $2\times 2$ matrices $A$ and $B$ with determinant $1$.

Consider the $2\times 2$ matrix

\[A=\begin{bmatrix}

\cos \theta & -\sin \theta\\

\sin \theta& \cos \theta \end{bmatrix},\]
where $\theta$ is a real number $0\leq \theta < 2\pi$.

**(a)** Find the characteristic polynomial of the matrix $A$.

**(b)** Find the eigenvalues of the matrix $A$.

**(c)** Determine the eigenvectors corresponding to each of the eigenvalues of $A$.

Consider the complex matrix

\[A=\begin{bmatrix}

\sqrt{2}\cos x & i \sin x & 0 \\

i \sin x &0 &-i \sin x \\

0 & -i \sin x & -\sqrt{2} \cos x

\end{bmatrix},\]
where $x$ is a real number between $0$ and $2\pi$.

Determine for which values of $x$ the matrix $A$ is diagonalizable.

When $A$ is diagonalizable, find a diagonal matrix $D$ so that $P^{-1}AP=D$ for some nonsingular matrix $P$.

Let $A$ be a square matrix.

Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues of $A$.

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.

Let $A$ be a square matrix and its characteristic polynomial is give by

\[p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).\]
Find the rank of $A$.

(The Ohio State University, Linear Algebra Final Exam Problem)

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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*)

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Let

\[A=\begin{bmatrix}

1 & 2\\

4& 3

\end{bmatrix}.\]

**(a)** Find eigenvalues of the matrix $A$.

**(b)** Find eigenvectors for each eigenvalue of $A$.

**(c)** Diagonalize the matrix $A$. That is, find an invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

**(d)** Diagonalize the matrix $A^3-5A^2+3A+I$, where $I$ is the $2\times 2$ identity matrix.

**(e)** Calculate $A^{100}$. (You do not have to compute $5^{100}$.)

**(f)** Calculate

\[(A^3-5A^2+3A+I)^{100}.\]
Let $w=2^{100}$. Express the solution in terms of $w$.

Let

\[A=\begin{bmatrix}

1-a & a\\

-a& 1+a

\end{bmatrix}\]
be a $2\times 2$ matrix, where $a$ is a complex number.

Determine the values of $a$ such that the matrix $A$ is diagonalizable.

(*Nagoya University, Linear Algebra Exam Problem*)

Determine whether the matrix

\[A=\begin{bmatrix}

0 & 1 & 0 \\

-1 &0 &0 \\

0 & 0 & 2

\end{bmatrix}\]
is diagonalizable.

If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

Add to solve laterDetermine whether each of the following statements is True or False.

**(a)** If $A$ and $B$ are $n \times n$ matrices, and $P$ is an invertible $n \times n$ matrix such that $A=PBP^{-1}$, then $\det(A)=\det(B)$.

**(b)** If the characteristic polynomial of an $n \times n$ matrix $A$ is

\[p(\lambda)=(\lambda-1)^n+2,\]
then $A$ is invertible.

**(c)** If $A^2$ is an invertible $n\times n$ matrix, then $A^3$ is also invertible.

**(d)** If $A$ is a $3\times 3$ matrix such that $\det(A)=7$, then $\det(2A^{\trans}A^{-1})=2$.

**(e)** If $\mathbf{v}$ is an eigenvector of an $n \times n$ matrix $A$ with corresponding eigenvalue $\lambda_1$, and if $\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_2$, then $\mathbf{v}+\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_1+\lambda_2$.

(Stanford University, Linear Algebra Exam Problem)

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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.

**(a)** Let $A$ be a real orthogonal $n\times n$ matrix. Prove that the length (magnitude) of each eigenvalue of $A$ is $1$

**(b)** Let $A$ be a real orthogonal $3\times 3$ matrix and suppose that the determinant of $A$ is $1$. Then prove that $A$ has $1$ as an eigenvalue.

Let $n$ be an odd integer and let $A$ be an $n\times n$ real matrix.

Prove that the matrix $A$ has at least one real eigenvalue.

Let $A$ be an $n\times n$ real matrix.

Prove that if $\lambda$ is an eigenvalue of $A$, then its complex conjugate $\bar{\lambda}$ is also an eigenvalue of $A$.

Add to solve later**(a)** Is the matrix $A=\begin{bmatrix}

1 & 2\\

0& 3

\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}

3 & 0\\

1& 2

\end{bmatrix}$?

**(b)** Is the matrix $A=\begin{bmatrix}

0 & 1\\

5& 3

\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}

1 & 2\\

4& 3

\end{bmatrix}$?

**(c)** Is the matrix $A=\begin{bmatrix}

-1 & 6\\

-2& 6

\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}

3 & 0\\

0& 2

\end{bmatrix}$?

**(d)** Is the matrix $A=\begin{bmatrix}

-1 & 6\\

-2& 6

\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}

1 & 2\\

-1& 4

\end{bmatrix}$?

Prove that if $A$ and $B$ are similar matrices, then their determinants are the same.

Add to solve laterLet $A$ be $n\times n$ matrix and let $\lambda_1, \lambda_2, \dots, \lambda_n$ be all the eigenvalues of $A$. (Some of them may be the same.)

For each positive integer $k$, prove that $\lambda_1^k, \lambda_2^k, \dots, \lambda_n^k$ are all the eigenvalues of $A^k$.

Add to solve laterLet $A$ be an $n\times n$ matrix. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities.

What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix?

(*The Ohio State University, Linear Algebra Final Exam Problem*)

Let $A$ be an $n\times n$ matrix with the characteristic polynomial

\[p(t)=t^3(t-1)^2(t-2)^5(t+2)^4.\]
Assume that the matrix $A$ is diagonalizable.

**(a)** Find the size of the matrix $A$.

**(b)** Find the dimension of the eigenspace $E_2$ corresponding to the eigenvalue $\lambda=2$.

**(c)** Find the nullity of $A$.

(The Ohio State University, Linear Algebra final exam problem)

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