## Example 1

Let $$A = \begin{bmatrix} 1 & 3 & 4\\ -5 & 2 & 0\\ -1 & 2 & 1\end{bmatrix}$$. Let $$\sigma$$ denote the permutation $$\begin{pmatrix} 1 & 2 & 3\\ 3 & 1 & 2 \end{pmatrix}$$. What is $$A_{1,\sigma(1)} A_{2,\sigma(2)} A_{3,\sigma(3)}$$?

## Example 2

Compute the determinant of $$\begin{bmatrix} -i & 1 & 0 \\ 1 & 2 & 0 \\ -1 & 1 & 1\end{bmatrix}$$.

## Example 3

Consider the matrix $$A =\left[ \begin{array}{cccc} k & 1 & 1 & 1 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & k & 1 \\ 1 & -1 & 2 & -1 \end{array} \right].$$ Determine all values of $$k$$ such that $$A$$ is singular.

## Example 4

Let $$A$$ be an $$n\times n$$ matrix over some field. Let $$B = \begin{bmatrix} C_{1,1} & C_{2,1} & \cdots & C_{n,1} \\ C_{1,2} & C_{2,2} & \cdots & C_{n,2} \\ \vdots & \vdots & \ddots & \vdots \\ C_{1,n} & C_{2,n} & \cdots & C_{n,n}\end{bmatrix}$$ where $$C_{i,j} = (-1)^{i+j}A(i\mid j)$$. Verify that $$AB = \det(A)I_n$$.

## Example 5

Let $$A,B \in \mathbb{C}^{3\times 3}$$. Suppose that $$\det(A) = 1-i$$ and $$B = \begin{bmatrix} 1 & 3 & 4\\ 0 & 2 & 5\\ 0 & 0 & i\end{bmatrix}$$. What is $$\det(A^{-2}B^\mathsf{T})$$?