Let \[\mathbf{A}=\begin{pmatrix} 1 & 2 & 0 & -1 & 5 \\ 2 & 0 & 2 & 0 & 1 \\ 1 & 1 & -1 & 3 & 2 \\ 0 & 3 & -3 & 2 & 6 \end{pmatrix}.\] Compute \(\operatorname{Ker}f_\mathbf{A}\) and \(\operatorname{Im}f_\mathbf{A}\) using Gauss elimination.

Compute \(\mathbf{A}^{-1}\) and \(\mathbf{B}^{-1}\) (if they exist) for \[\mathbf{A}=\begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}, \qquad \mathbf{B}=\begin{pmatrix} 2 & - 1 & -1 \\ - 1 & 2 & -1 \\ -1 & -1 & 2 \end{pmatrix}.\]

Show that \(\mathbf{A}\in M_{m,n}(\mathbb{K})\) has rank \(1\) if and only if \(\mathbf{A}=\vec{x}\vec{\xi}\) for some nonzero column vector \(\vec{x} \in \mathbb{K}^m\) and nonzero row vector \(\vec{\xi} \in \mathbb{K}_n.\)

Let \(\mathbf{A}\in M_{m,n}(\mathbb{K})\) with \(m<n.\) Show that \(\mathbf{A}\) does not admit a left inverse.