1.
Construct a $3 \times 4$ matrix whose elements are given by $a_{ij} = \frac{1}{2}|-3i + j|$.
2.
Find the total number of possible matrices of order $3 \times 3$ with each entry being $0$ or $1$.
3.
If a matrix has 24 elements, list all the possible orders it can have. What if it has 13 elements?
4.
Construct a $2 \times 2$ matrix $A = [a_{ij}]$ whose elements are defined by $a_{ij} = e^{2ix} \sin(jx)$.
5.
Prove that the matrix $A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$ is an idempotent matrix.
6.
Show that $B = \begin{bmatrix} 1 & -3 & -4 \\ -1 & 3 & 4 \\ 1 & -3 & -4 \end{bmatrix}$ is nilpotent of index 2.
7.
Find $x, y, z$ if the matrix $A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix}$ is an orthogonal matrix (i.e., $AA' = I$).
8.
If $A$ is a square matrix and $k$ is a scalar, express the trace of $kA$ in terms of the trace of $A$.
9.
Find $x, y, a, b$ if $\begin{bmatrix} 2x-3y & a-b \\ 2a+b & x+4y \end{bmatrix} = \begin{bmatrix} 1 & -2 \\ 5 & 6 \end{bmatrix}$.
10.
Find $x$ and $y$ if $2 \begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix} + \begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix}$.
11.
Solve for $x, y, z,$ and $t$: $2\begin{bmatrix} x & z \\ y & t \end{bmatrix} + 3\begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix} = 3\begin{bmatrix} 3 & 5 \\ 4 & 6 \end{bmatrix}$.
12.
If $A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$, $B = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}$, and $C = \begin{bmatrix} 1 & 1 \\ 2 & 3 \end{bmatrix}$, compute $(AB)C$ and verify that it equals $A(BC)$.
13.
Let $f(x) = x^2 - 5x + 6$. Find $f(A)$ if $A = \begin{bmatrix} 2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0 \end{bmatrix}$.
14.
Find a matrix $X$ such that $X \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} = \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix}$.
15.
If $A = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$, show that $A^2 = \begin{bmatrix} \cos 2\alpha & -\sin 2\alpha \\ \sin 2\alpha & \cos 2\alpha \end{bmatrix}$.
16.
Find $x$ if $\begin{bmatrix} 1 & x & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ x \end{bmatrix} = O$.
17.
Verify that $(AB)' = B'A'$ for $A = \begin{bmatrix} 1 \\ -4 \\ 3 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 2 & 1 \end{bmatrix}$.
18.
If $A = \begin{bmatrix} \sin \alpha & \cos \alpha \\ -\cos \alpha & \sin \alpha \end{bmatrix}$, verify that $A'A = I$.
19.
For any square matrix $A$ with real numbers, prove that $(A+A')$ is a symmetric matrix.
20.
Using the properties of transpose, prove that $(A^n)' = (A')^n$ for any positive integer $n$.
21.
Express the matrix $B = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.
22.
Show that all positive odd integral powers of a skew-symmetric matrix are skew-symmetric.
23.
Let $A$ and $B$ be symmetric matrices. Prove that $AB$ is symmetric if and only if $A$ and $B$ commute (i.e., $AB = BA$).
24.
If $A$ and $B$ are symmetric matrices of the same order, prove that $AB + BA$ is a symmetric matrix.
25.
Using elementary transformations, find the inverse of the matrix $A = \begin{bmatrix} 1 & 2 \\ 2 & -1 \end{bmatrix}$.
26.
Obtain the inverse of the matrix $A = \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}$ using elementary row operations.
27.
By using elementary column operations, find the inverse of $A = \begin{bmatrix} 3 & -1 \\ -4 & 2 \end{bmatrix}$.
28.
Prove that the inverse of an invertible symmetric matrix is a symmetric matrix.
