## Example 1

Find the inverse matrix of $$\begin{bmatrix} -i & 1 \\ 2 & 0\end{bmatrix}$$.

## Example 2

Let $$A = \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 0\end{bmatrix}$$ be defined over $$GF(2)$$.

1. Find $$A^{-1}$$.

2. Write $$A$$ as a product of elementary matrices.

## Example 3

Show that $$A = \begin{bmatrix} 4 & -1 & 2\\ -2 & 1 & 1\\ 2 & 0 & 3\end{bmatrix}$$ is singular.

## Example 4

Simplify the following matrix expression: $$\left(\begin{bmatrix} 1 \\ 2 \end{bmatrix} \begin{bmatrix} 1 & -1 \end{bmatrix} - \begin{bmatrix} 2 & -1 \\ 0 & 1\end{bmatrix}^3\right)^\mathsf{T}.$$

## Example 5

Let $$A = \begin{bmatrix} a & b \\ c & d\end{bmatrix}$$ be defined over the real numbers. Prove that $$A$$ is invertible if and only if $$ad - bc \neq 0$$.

## Example 6

Let $$\mathbb{F}$$ denote a field. Let $$A \in \mathbb{F}^{m\times n}$$. Let $$\lambda \in \mathbb{F}$$. Prove that $$(\lambda A)^\mathsf{T} = \lambda A^\mathsf{T}$$.

## Example 7

Let $$A \in \mathbb{F}^{n\times n}$$ where $$n$$ is a positive integer and $$\mathbb{F}$$ denotes a field. Prove that if $$A$$ is nonsingular, then $$A$$ is invertible.