EITHER

It is given that \(1\) and \(4\) are eigenvalues of the matrix \(\mathbf A\), where

\(\mathbf A=\begin{pmatrix}1&-3&-3\\-8&6&-3\\8&-2&7\end{pmatrix}.\)

Find eigenvectors corresponding to each of these eigenvalues.

Given further that \(\begin{pmatrix}0\\1\\-1\end{pmatrix}\) is an eigenvector of \(\mathbf A\), find the corresponding eigenvalue.

Write down matrices \(\mathbf P\) and \(\mathbf D\) such that \(\mathbf P^{-1}\mathbf A\mathbf P=\mathbf D\), where \(\mathbf D\) is a diagonal matrix, and find \(\mathbf P^{-1}\).

Write down a matrix \(\mathbf C\) such that \(\mathbf C^2=\mathbf D\), and deduce a matrix \(\mathbf B\) such that \(\mathbf B^2=\mathbf A\).