# Tagged: matrix multiplication

## Problem 641

Let $\mathbf{v} = \begin{bmatrix} 2 & -5 & -1 \end{bmatrix}$.

Find all $3 \times 1$ column vectors $\mathbf{w}$ such that $\mathbf{v} \mathbf{w} = 0$.

## Problem 492

Let
$D=\begin{bmatrix} d_1 & 0 & \dots & 0 \\ 0 &d_2 & \dots & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \dots & d_n \end{bmatrix}$ be a diagonal matrix with distinct diagonal entries: $d_i\neq d_j$ if $i\neq j$.
Let $A=(a_{ij})$ be an $n\times n$ matrix such that $A$ commutes with $D$, that is,
$AD=DA.$ Then prove that $A$ is a diagonal matrix.

## Problem 383

Let
$A=\begin{bmatrix} 1 & 1 & 1 \\ 0 &0 &1 \\ 0 & 0 & 1 \end{bmatrix}$ be a $3\times 3$ matrix. Then find the formula for $A^n$ for any positive integer $n$.

## Problem 287

Let $V$ be the vector space of all $3\times 3$ real matrices.
Let $A$ be the matrix given below and we define
$W=\{M\in V \mid AM=MA\}.$ That is, $W$ consists of matrices that commute with $A$.
Then $W$ is a subspace of $V$.

Determine which matrices are in the subspace $W$ and find the dimension of $W$.

(a) $A=\begin{bmatrix} a & 0 & 0 \\ 0 &b &0 \\ 0 & 0 & c \end{bmatrix},$ where $a, b, c$ are distinct real numbers.

(b) $A=\begin{bmatrix} a & 0 & 0 \\ 0 &a &0 \\ 0 & 0 & b \end{bmatrix},$ where $a, b$ are distinct real numbers.

## Problem 263

Let $A$ and $B$ are matrices such that the matrix product $AB$ is defined and $AB$ is a square matrix.
Is it true that the matrix product $BA$ is also defined and $BA$ is a square matrix? If it is true, then prove it. If not, find a counterexample.

## Problem 155

Let $A$ be an $m \times n$ matrix.
Let $\calN(A)$ be the null space of $A$. Suppose that $\mathbf{u} \in \calN(A)$ and $\mathbf{v} \in \calN(A)$.
Let $\mathbf{w}=3\mathbf{u}-5\mathbf{v}$.

Then find $A\mathbf{w}$.

## Problem 111

Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings.

(a) The product $AB$ is symmetric if and only if $AB=BA$.

(b) If the product $AB$ is a diagonal matrix, then $AB=BA$.

## Problem 98

Let $A$ and $B$ be $n\times n$ matrices. Suppose that the matrix product $AB=O$, where $O$ is the $n\times n$ zero matrix.

Is it true that the matrix product with opposite order $BA$ is also the zero matrix?
If so, give a proof. If not, give a counterexample.

## Problem 96

Let $A$ and $B$ be $2\times 2$ matrices.

Prove or find a counterexample for the statement that $(A-B)(A+B)=A^2-B^2$.

## Problem 19

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 $\tr(A^2)>0$.