The square matrix \(\mathbf{A}\) has an eigenvalue \(\lambda\) with corresponding eigenvector \(\mathbf{e}\). The non-singular matrix \(\mathbf{M}\) is of the same order as \(\mathbf{A}\). Show that \(\mathbf{M e}\) is an eigenvector of the matrix \(\mathbf{B}\), where \(\mathbf{B}=\mathbf{M} \mathbf{A} \mathbf{M}^{-1}\), and that \(\lambda\) is the corresponding eigenvalue.

Let
\(\mathbf{A}=\left(\begin{array}{rrr} -1 & 2 & 1 \\ 0 & 1 & 4 \\ 0 & 0 & 2 \end{array}\right)\)

Write down the eigenvalues of \(\mathbf{A}\) and obtain corresponding eigenvectors.

Given that
\(\mathbf{M}=\left(\begin{array}{lll} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)\)
find the eigenvalues and corresponding eigenvectors of \(\mathbf{B}\).