Compute the determinant of the following matrices: \[\begin{pmatrix} 1 & \mathrm{i}\\ 2-\mathrm{i}& 3 \end{pmatrix}, \quad \begin{pmatrix}2 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 2 \end{pmatrix},\quad \begin{pmatrix} 1 & 0 & 0 & 0 \\ 5 & 2 & 0 & 0 \\ 8 & 6 & 3 & 0 \\ 0 & 9 & 7 & 4\end{pmatrix}, \quad \begin{pmatrix} 1 & 4 & 1 & 3 \\ 2 & 3 & 5 & 0 \\ 4 & 1 & 0 & 0 \\ 2 & 0 & 0 & 0 \end{pmatrix}.\]

Use induction to compute the determinant of the following matrices: \[\begin{pmatrix} &&&&& 1 \\ &&&& 1 & \\ &&& 1 && \\ && \ddots &&& \\ & 1 &&&& \\ 1 &&&&&\end{pmatrix}, \qquad \begin{pmatrix} 2 & -1 &&&& \\ -1 & 2 & -1 &&& \\ & -1 & 2 & -1 && \\ && -1 & 2 &&\\ &&&& \ddots & \\ &&&&& 2 & -1 \\ &&&&& -1 & 2\end{pmatrix}.\] Here the entries that are not printed are zero.

Let \(n \in \mathbb{N}\) and \(\mathbf{A}\in M_{n,n}(\mathbb{K}).\) Show that \(\det(\mathbf{A})=\det\left(\mathbf{A}^T\right).\)

Consider a \(2n\times 2n\)-matrix of the form \[\mathbf{M}=\begin{pmatrix} \mathbf{A}& \mathbf{B}\\ \mathbf{0}_n & \mathbf{D}\end{pmatrix},\] where \(\mathbf{A}\) and \(\mathbf{D}\) are square submatrices of size \(n.\) Show that \(\det(\mathbf{M})=\det(\mathbf{A})\det(\mathbf{D}).\)

(\(\heartsuit\)). Consider a \(2n\times 2n\)-matrix of the form \[\mathbf{M}=\begin{pmatrix} \mathbf{A}& \mathbf{B}\\ \mathbf{C}& \mathbf{D}\end{pmatrix},\] where \(\mathbf{A},\mathbf{B},\mathbf{C},\mathbf{D}\) are square submatrices of size \(n.\) Show that \(\det(\mathbf{M})=\det(\mathbf{A}\mathbf{D}-\mathbf{C}\mathbf{B}),\) provided \(\mathbf{A}\mathbf{C}=\mathbf{C}\mathbf{A}.\) Show by example that the formula is false in general.