Due date: Monday 25. September 2023, 10 AM.
Exercise 1
Let \(n \in \mathbb{N}\) be a natural number and \(\mathbf{A}\in M_{n,n}(\mathbb{K}).\) Show that if \(\mathbf{A}\) has a row all of whose entries are zero, then \(\mathbf{A}\) does not admit an inverse.
Exercise 2
Let \(n \in \mathbb{N}\) be a natural number. A matrix \(\mathbf{A}\in M_{n,n}(\mathbb{K})\) is called nilpotent if there exists a natural number \(N>0\) so that \[\mathbf{A}^N=\underbrace{\mathbf{A}\mathbf{A}\cdots \mathbf{A}}_{N\text{-times}}=\mathbf{0}_n.\] Suppose \(\mathbf{A}\) is nilpotent. Show that \(\mathbf{B}=\mathbf{A}+\mathbf{1}_{n}\) admits an inverse matrix by computing an explicit formula for \(\mathbf{B}^{-1}.\)
Exercise 3
Let \[\mathbf{A}=\begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{pmatrix}.\] Find a formula for \(\mathbf{A}^n,\) where \(n \in \mathbb{N}\) and verify your formula by induction.
Exercise 4
Show that all \(2\)-by-\(2\) matrices \(\mathbf{A}\) with real entries so that \(\mathbf{A}\mathbf{A}^T=s \mathbf{1}_{2}\) for some scalar \(s\) are of the form \[\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\qquad \text{or} \qquad \begin{pmatrix} a & b \\ b & -a \end{pmatrix}\] for some scalars \(a,b.\)
Let \(V\) denote the set of such matrices of the first type and let \(\mathbf{A}\in V.\) Give a geometric interpretation of the mapping \(f_\mathbf{A}: \mathbb{R}^2 \to \mathbb{R}^2.\)
Conclude, without calculation, that for \(\mathbf{A},\mathbf{B}\in V,\) we have \(\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}.\)
Exercise 5
(\(\heartsuit\)). How does \(V\) of Exercise 4 relate to complex numbers?