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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 \(\det(A^{-2}B^\mathsf{T})\)?