For the matrix:
\begin{equation*}
A = \begin{pmatrix} -1 & 1 & 3 \\ 1 & 2 & 0 \\ 3 & 0 & 2 \end{pmatrix},
\end{equation*}
show that the eigenvalues are 2, 4, and -3.
Evaluate Trace$[A]$ and Det$[A]$. Explain how these results are (or are not) consistent with what you found for the
eigenvalues of A.
Find the normalised eigenvectors of A.
Use the normalised eigenvectors of part iii) to construct the matrix $U$:
\begin{equation*}
U = (\vec{x}^{\lambda_1}, \vec{x}^{\lambda_2}, \vec{x}^{\lambda_3}),
\end{equation*}
where $\vec{x}^{\lambda_i}$ is the (column) eigenvector corresponding to the eigenvalue $\lambda = \lambda_i$, $i = 1, 2, 3$.
Show that
\begin{equation*}
U^{-1}AU = \begin{pmatrix} \lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & \lambda_3 \end{pmatrix}.
\end{equation*}
