# Tagged: matrix operation

## Problem 635

Let $A$ and $B$ be $n \times n$ matrices, and $\mathbf{v}$ an $n \times 1$ column vector.

Use the matrix components to prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$.

## 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 300

Let $A$ be the coefficient matrix of the system of linear equations
\begin{align*}
-x_1-2x_2&=1\\
2x_1+3x_2&=-1.
\end{align*}

(a) Solve the system by finding the inverse matrix $A^{-1}$.

(b) Let $\mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ be the solution of the system obtained in part (a).
Calculate and simplify
$A^{2017}\mathbf{x}.$

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

## Problem 297

Let $A, B, C$ be the following $3\times 3$ matrices.
$A=\begin{bmatrix} 1 & 2 & 3 \\ 4 &5 &6 \\ 7 & 8 & 9 \end{bmatrix}, B=\begin{bmatrix} 1 & 0 & 1 \\ 0 &3 &0 \\ 1 & 0 & 5 \end{bmatrix}, C=\begin{bmatrix} -1 & 0\ & 1 \\ 0 &5 &6 \\ 3 & 0 & 1 \end{bmatrix}.$ Then compute and simplify the following expression.
$(A^{\trans}-B)^{\trans}+C(B^{-1}C)^{-1}.$

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

Let $A$ and $B$ are $n \times n$ matrices with real entries.
Assume that $A+B$ is invertible. Then show that
$A(A+B)^{-1}B=B(A+B)^{-1}A.$