# Tagged: matrix

## Problem 189

Let $C$ be a $4 \times 4$ matrix with all eigenvalues $\lambda=2, -1$ and eigensapces
$E_2=\Span\left \{\quad \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} \quad\right \} \text{ and } E_{-1}=\Span\left \{ \quad\begin{bmatrix} 1 \\ 2 \\ 1 \\ 1 \end{bmatrix},\quad \begin{bmatrix} 1 \\ 1 \\ 1 \\ 2 \end{bmatrix} \quad\right\}.$

Calculate $C^4 \mathbf{u}$ for $\mathbf{u}=\begin{bmatrix} 6 \\ 8 \\ 6 \\ 9 \end{bmatrix}$ if possible. Explain why if it is not possible!

(The Ohio State University Linear Algebra Exam Problem)

## Problem 186

Let $A$ and $B$ be $n\times n$ matrices, where $n$ is an integer greater than $1$.

Is it true that
$\det(A+B)=\det(A)+\det(B)?$ If so, then give a proof. If not, then give a counterexample.

## Problem 185

Let $A=(a_{ij})$ be an $n \times n$ matrix.
We say that $A=(a_{ij})$ is a right stochastic matrix if each entry $a_{ij}$ is nonnegative and the sum of the entries of each row is $1$. That is, we have
$a_{ij}\geq 0 \quad \text{ and } \quad a_{i1}+a_{i2}+\cdots+a_{in}=1$ for $1 \leq i, j \leq n$.

Let $A=(a_{ij})$ be an $n\times n$ right stochastic matrix. Then show the following statements.

(a)The stochastic matrix $A$ has an eigenvalue $1$.

(b) The absolute value of any eigenvalue of the stochastic matrix $A$ is less than or equal to $1$.

## Problem 184

Suppose that $A$ and $P$ are $3 \times 3$ matrices and $P$ is invertible matrix.
If
$P^{-1}AP=\begin{bmatrix} 1 & 2 & 3 \\ 0 &4 &5 \\ 0 & 0 & 6 \end{bmatrix},$ then find all the eigenvalues of the matrix $A^2$.

## Problem 183

Let $A$ be an $n \times n$ matrix. Suppose that the matrix $A^2$ has a real eigenvalue $\lambda>0$. Then show that either $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of the matrix $A$.

## Problem 181

Suppose that $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ is an eigenvector of a matrix $A$ corresponding to the eigenvalue $3$ and that $\begin{bmatrix} 2 \\ 1 \end{bmatrix}$ is an eigenvector of $A$ corresponding to the eigenvalue $-2$.
Compute $A^2\begin{bmatrix} 4 \\ 3 \end{bmatrix}$.

(Stanford University Linear Algebra Exam Problem)

## Problem 180

Suppose the following information is known about a $3\times 3$ matrix $A$.
$A\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}=6\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A\begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}=3\begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}, \quad A\begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix}=3\begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}.$

(a) Find the eigenvalues of $A$.

(b) Find the corresponding eigenspaces.

(c) In each of the following questions, you must give a correct reason (based on the theory of eigenvalues and eigenvectors) to get full credit.
Is $A$ a diagonalizable matrix?
Is $A$ an invertible matrix?
Is $A$ an idempotent matrix?

(Johns Hopkins University Linear Algebra Exam)

## Problem 178

Let
$\begin{bmatrix} 0 & 0 & 1 \\ 1 &0 &0 \\ 0 & 1 & 0 \end{bmatrix}.$

(a) Find the characteristic polynomial and all the eigenvalues (real and complex) of $A$. Is $A$ diagonalizable over the complex numbers?

(b) Calculate $A^{2009}$.

(Princeton University, Linear Algebra Exam)

## Problem 176

Let $A$ be an $n \times n$ matrix. We say that $A$ is idempotent if $A^2=A$.

(a) Find a nonzero, nonidentity idempotent matrix.

(b) Show that eigenvalues of an idempotent matrix $A$ is either $0$ or $1$.

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

## Problem 169

Find all the values of $x$ so that the following matrix $A$ is a singular matrix.
$A=\begin{bmatrix} x & x^2 & 1 \\ 2 &3 &1 \\ 0 & -1 & 1 \end{bmatrix}.$

## Problem 168

Let
$A=\begin{bmatrix} 1 & -x & 0 & 0 \\ 0 &1 & -x & 0 \\ 0 & 0 & 1 & -x \\ 0 & 1 & 0 & -1 \end{bmatrix}$ be a $4\times 4$ matrix. Find all values of $x$ so that the matrix $A$ is singular.

## Problem 166

Let $V$ be the vector space of all $2\times 2$ matrices. Let $W$ be a subset of $V$ consisting of all $2\times 2$ skew-symmetric matrices. (Recall that a matrix $A$ is skew-symmetric if $A^{\trans}=-A$.)

(a) Prove that the subset $W$ is a subspace of $V$.

(b) Find the dimension of $W$.

(The Ohio State University Linear Algebra Exam Problem)

## Problem 165

Let $P_2$ be the vector space of all polynomials of degree two or less.
Consider the subset in $P_2$
$Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},$ where
\begin{align*}
&p_1(x)=1, &p_2(x)=x^2+x+1, \\
&p_3(x)=2x^2, &p_4(x)=x^2-x+1.
\end{align*}

(a) Use the basis $B=\{1, x, x^2\}$ of $P_2$, give the coordinate vectors of the vectors in $Q$.

(b) Find a basis of the span $\Span(Q)$ consisting of vectors in $Q$.

(c) For each vector in $Q$ which is not a basis vector you obtained in (b), express the vector as a linear combination of basis vectors.

(The Ohio State University Linear Algebra Exam Problem)

## Problem 164

Let $T:\R^4 \to \R^3$ be a linear transformation defined by
$T\left (\, \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} \,\right) = \begin{bmatrix} x_1+2x_2+3x_3-x_4 \\ 3x_1+5x_2+8x_3-2x_4 \\ x_1+x_2+2x_3 \end{bmatrix}.$

(a) Find a matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$.

(b) Find a basis for the null space of $T$.

(c) Find the rank of the linear transformation $T$.

(The Ohio State University Linear Algebra Exam Problem)

## Problem 152

Let $V$ be the vector space of all $2\times 2$ matrices, and let the subset $S$ of $V$ be defined by $S=\{A_1, A_2, A_3, A_4\}$, where
\begin{align*}
A_1=\begin{bmatrix}
1 & 2 \\
-1 & 3
A_2=\begin{bmatrix}
0 & -1 \\
1 & 4
A_3=\begin{bmatrix}
-1 & 0 \\
1 & -10
A_4=\begin{bmatrix}
3 & 7 \\
-2 & 6
\end{bmatrix}.
\end{align*}
Find a basis of the span $\Span(S)$ consisting of vectors in $S$ and find the dimension of $\Span(S)$.

## Problem 150

Show that the set
$S=\{1, 1-x, 3+4x+x^2\}$ is a basis of the vector space $P_2$ of all polynomials of degree $2$ or less.

## Problem 143

Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$.

(a) The set $S$ consisting of all $n\times n$ symmetric matrices.

(b) The set $T$ consisting of all $n \times n$ skew-symmetric matrices.

(c) The set $U$ consisting of all $n\times n$ nonsingular matrices.

## Problem 140

Let $A$ be an $m\times n$ matrix. The nullspace of $A$ is denoted by $\calN(A)$.
The dimension of the nullspace of $A$ is called the nullity of $A$.
Prove the followings.

(a) $\calN(A)=\calN(A^{\trans}A)$.

(b) $\rk(A)=\rk(A^{\trans}A)$.

## Problem 139

Let $A_1, A_2, \dots, A_m$ be $n\times n$ Hermitian matrices. Show that if
$A_1^2+A_2^2+\cdots+A_m^2=\calO,$ where $\calO$ is the $n \times n$ zero matrix, then we have $A_i=\calO$ for each $i=1,2, \dots, m$.

## Problem 138

Find the determinant of the matix
$A=\begin{bmatrix} 100 & 101 & 102 \\ 101 &102 &103 \\ 102 & 103 & 104 \end{bmatrix}.$