## Calculate $A^{10}$ for a Given Matrix $A$

## 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*)

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*)

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*)

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

Then the* kernel* of $A$ is defined as $\ker(A)=\{ x\in \R^n \mid Ax=0 \}$.

The kernel is also called the* null space* of $A$.

Suppose that $A$ is an $m \times n$ real matrix such that $\ker(A)=0$. Prove that $A^{\trans}A$ is invertible.

(*Stanford University Linear Algebra Exam*)

Suppose that $A$ is a diagonalizable $n\times n$ matrix and has only $1$ and $-1$ as eigenvalues.

Show that $A^2=I_n$, where $I_n$ is the $n\times n$ identity matrix.

(*Stanford University Linear Algebra Exam*)

See below for a generalized problem.

Add to solve laterLet $A$ be an $n$ by $n$ matrix with entries in complex numbers $\C$. Its only eigenvalues are $1,2,3,4,5$, possibly with multiplicities. What is the rank of the matrix $A+I_n$, where $I_n$ is the identity $n$ by $n$ matrix.

(*UCB-University of California, Berkeley, Exam*)

**(a)** Let

\[A=\begin{bmatrix}

a_{11} & a_{12}\\

a_{21}& a_{22}

\end{bmatrix}\]
be a matrix such that $a_{11}+a_{12}=1$ and $a_{21}+a_{22}=1$. Namely, the sum of the entries in each row is $1$.

(Such a matrix is called (right) * stochastic matrix* (also termed probability matrix, transition matrix, substitution matrix, or Markov matrix).)

Then prove that the matrix $A$ has an eigenvalue $1$.

**(b)** Find all the eigenvalues of the matrix

\[B=\begin{bmatrix}

0.3 & 0.7\\

0.6& 0.4

\end{bmatrix}.\]

**(c)** For each eigenvalue of $B$, find the corresponding eigenvectors.

Suppose that $S$ is a fixed invertible $3$ by $3$ matrix. This question is about all the matrices $A$ that are diagonalized by $S$, so that $S^{-1}AS$ is diagonal. Show that these matrices $A$ form a subspace of $3$ by $3$ matrix space.

(*MIT-Massachusetts Institute of Technology Exam*)

Let

\[ A=\begin{bmatrix}

2 & 0 & 10 \\

0 &7+x &-3 \\

0 & 4 & x

\end{bmatrix}.\]
Find all values of $x$ such that $A$ is invertible.

(*Stanford University Linear Algebra Exam*)

In this problem, we will show that the concept of non-singularity of a matrix is equivalent to the concept of invertibility.

That is, we will prove that:

A matrix $A$ is nonsingular if and only if $A$ is invertible.

**(a)** Show that if $A$ is invertible, then $A$ is nonsingular.

**(b)** Let $A, B, C$ be $n\times n$ matrices such that $AB=C$.

Prove that if either $A$ or $B$ is singular, then so is $C$.

**(c)** Show that if $A$ is nonsingular, then $A$ is invertible.

An $n \times n$ matrix $A$ is called ** nonsingular** if the only solution of the equation $A \mathbf{x}=\mathbf{0}$ is the zero vector $\mathbf{x}=\mathbf{0}$.

Otherwise $A$ is called

**(a)** Show that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is also nonsingular.

**(b)** Show that if $A$ is nonsingular, then the column vectors of $A$ are linearly independent.

**(c)** Show that an $n \times n$ matrix $A$ is nonsingular if and only if the equation $A\mathbf{x}=\mathbf{b}$ has a unique solution for any vector $\mathbf{b}\in \R^n$.

**Restriction**

Do not use the fact that a matrix is nonsingular if and only if the matrix is invertible.

Find all eigenvalues of the following $n \times n$ matrix.

\[

A=\begin{bmatrix}

0 & 0 & \cdots & 0 &1 \\

1 & 0 & \cdots & 0 & 0\\

0 & 1 & \cdots & 0 &0\\

\vdots & \vdots & \ddots & \ddots & \vdots \\

0 & 0&\cdots & 1& 0 \\

\end{bmatrix}

\]

Let $A$ be an $n \times n$ matrix such that $\tr(A^n)=0$ for all $n \in \N$.

Then prove that $A$ is a nilpotent matrix. Namely there exist a positive integer $m$ such that $A^m$ is the zero matrix.

Let $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$. Then answer the following questions about the trace of a matrix.

**(a)** Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of $B$.

**(b)** Show that $\tr(AA^{\trans})$ is the sum of the square of the entries of $A$.

**(c)** Show that if $A$ is nonzero symmetric matrix, then $\tr(A^2)>0$.

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

Suppose that these matrices have a common eigenvector $\mathbf{x}$.

Show that $\det(AB-BA)=0$.

Read solution

Let $A$ be an $n \times n$ real matrix. Prove the followings.

**(a)** The matrix $AA^{\trans}$ is a symmetric matrix.

**(b) **The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal.

**(c)** The matrix $AA^{\trans}$ is non-negative definite.

(An $n\times n$ matrix $B$ is called *non-negative definite* if for any $n$ dimensional vector $\mathbf{x}$, we have $\mathbf{x}^{\trans}B \mathbf{x} \geq 0$.)

**(d)** All the eigenvalues of $AA^{\trans}$ is non-negative.

Let $A$ be an $n\times n$ matrix and let $\lambda_1, \dots, \lambda_n$ be its eigenvalues.

Show that

**(1) ** $$\det(A)=\prod_{i=1}^n \lambda_i$$

**(2)** $$\tr(A)=\sum_{i=1}^n \lambda_i$$

Here $\det(A)$ is the determinant of the matrix $A$ and $\tr(A)$ is the trace of the matrix $A$.

Namely, prove that (1) the determinant of $A$ is the product of its eigenvalues, and (2) the trace of $A$ is the sum of the eigenvalues.

Read solution

Let $A= \begin{bmatrix}

1 & 2\\

2& 1

\end{bmatrix}$.

Compute $A^n$ for any $n \in \N$.

Let $A=\begin{bmatrix}

a & 0\\

0& b

\end{bmatrix}$.

Show that

**(1)** $A^n=\begin{bmatrix}

a^n & 0\\

0& b^n

\end{bmatrix}$ for any $n \in \N$.

**(2) **Let $B=S^{-1}AS$, where $S$ be an invertible $2 \times 2$ matrix.

Show that $B^n=S^{-1}A^n S$ for any $n \in \N$

Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same.

Add to solve later