1.
Let $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and let $P(x) = x^2 - 5x - 2$. Show that $P(A) = O$. Hence, use this result to compute $A^3$.
2.
If $A$ and $B$ are skew-symmetric matrices of the same order, prove that the matrix $AB$ is symmetric if and only if $A$ and $B$ commute (i.e., $AB = BA$).
3.
A square matrix $A$ is said to be orthogonal if $AA' = A'A = I$. Prove that if $A$ is an orthogonal matrix, then $A^{-1} = A'$.
4.
Let $A = \begin{bmatrix} \cos \theta & i\sin \theta \\ i\sin \theta & \cos \theta \end{bmatrix}$. Prove by the Principle of Mathematical Induction that $A^n = \begin{bmatrix} \cos n\theta & i\sin n\theta \\ i\sin n\theta & \cos n\theta \end{bmatrix}$ for all $n \in \mathbb{N}$, where $i^2 = -1$.
5.
Let $A$ and $B$ be symmetric matrices of order $n$. Prove that the expression $A(BA) + B(AB)$ yields a symmetric matrix, while $A(BA) - B(AB)$ yields a skew-symmetric matrix.
6.
Let $A$ be a nilpotent matrix of index 2 (meaning $A^2 = O$). Prove that the inverse of the matrix $(I - A)$ is given by $(I + A)$.
7.
If $A$ is an idempotent matrix (i.e., $A^2 = A$), compute the simplified value of $(I+A)^3 - 7A$.
8.
Find the matrix $A$ if it satisfies the equation: $\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix} A \begin{bmatrix} -3 & 2 \\ 5 & -3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.
9.
If $A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$, find all possible real values of $\alpha$ and $\beta$ such that $(\alpha I + \beta A)^2 = A$.
10.
Prove that if $A$ and $B$ are square matrices such that $AB = BA$, then for any positive integer $n \ge 1$, $AB^n = B^nA$.
11.
Prove that for any two matrices $A$ and $B$ of order $n \times n$, $\text{Trace}(AB) = \text{Trace}(BA)$. Use this property to show that there exist no matrices $A$ and $B$ such that $AB - BA = I$.
12.
Let $A = \begin{bmatrix} 1 & a \\ 0 & 1 \end{bmatrix}$. Find a general formula for $A^n$, and then evaluate the matrix limit $\lim_{n \to \infty} \frac{1}{n} A^n$.
13.
If $f(x) = x^2 - 4x + 7$, and $A = \begin{bmatrix} 2 & 3 \\ -1 & 2 \end{bmatrix}$, show that $f(A) = O$. Using this polynomial equation, compute the exact matrix for $A^5$ without repeatedly multiplying matrices.
14.
A matrix $X$ has $(a+b)$ rows and $(a+2)$ columns, while the matrix $Y$ has $(b+1)$ rows and $(a+3)$ columns. If both the products $XY$ and $YX$ are well-defined, determine the values of $a$ and $b$.
15.
Let $A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & 1 & 1 \end{bmatrix}$. Find the inverse of matrix $A$ strictly by using elementary row transformations.
