Linear Algebra Matrix Exercises

• Linear Algebra Exercises

Exercise 1

• Given two matrices: $$ A=\begin{bmatrix} 1 & 2 & 3 \ 2 & 1 & 3 \ 1 & 1 & 2 \end{bmatrix} \qquad B=\begin{bmatrix} 2 & 1 & 0 \ 3 & 1 & 0 \ 0 & 0 & 2 \end{bmatrix} $$
• Calculate the matrix product $AB$.
• Calculate the matrix product $BA$.

Exercise 2

• Given the matrix: $$ A=\begin{bmatrix} \cos\theta & -\sin\theta \ \sin\theta & \cos\theta \end{bmatrix} $$
• Calculate $A^2$.
• Calculate $A^3$.
• Calculate $A^n$ for $n \ge 1$.

Exercise 3

• Given the matrix: $$ A=\begin{bmatrix} 2 & -1 \
• 1 & 2 \end{bmatrix} $$
• Show that $A^2 - 4A + 3I = 0$, where $I$ is the identity matrix and $0$ is the zero matrix.

Exercise 4

• Given the matrix: $$ A=\begin{bmatrix} 1 & -1 \ 2 & -1 \end{bmatrix} $$
• Calculate $A^2$.
• Calculate $A^3$.
• Calculate $A^n$ for $n \ge 1$.

Exercise 5

• Given the matrix: $$ A=\begin{bmatrix} 1 & 1 & 0 \ 0 & 1 & 1 \ 0 & 0 & 1 \end{bmatrix} $$
• Calculate $A^n$ for $n \ge 1$.

Exercise 6

• Given the matrix: $$ A=\begin{bmatrix} 1 & 2 & 3 \ 0 & 2 & 3 \ 0 & 0 & 1 \end{bmatrix} $$
• Calculate $A^n$ for $n \ge 1$.

Exercise 7

• Find all $2 \times 2$ matrices that commute with the matrix: $$ A=\begin{bmatrix} 1 & 0 \ 0 & -1 \end{bmatrix} $$

Exercise 8

• Find all $2 \times 2$ matrices that commute with the matrix: $$ A=\begin{bmatrix} 0 & 1 \
• 1 & 0 \end{bmatrix} $$