Question

Suppose that \(A\) is a \(3 \times 3\) matrix with eigenvalues \(\lambda_{1}=-1, \lambda_{2}=0\) and \(\lambda_{3}=1\), and corresponding eigenvectors
\[
\vec{v}_{1}=\left(\begin{array}{l}
1 \\
0 \\
2
\end{array}\right), \quad \vec{v}_{2}=\left(\begin{array}{r}
-1 \\
1 \\
0
\end{array}\right), \quad \vec{v}_{3}=\left(\begin{array}{l}
0 \\
0 \\
1
\end{array}\right)
\]
a) Find the matrix \(A\).
b) Compute the matrix \(A^{20}\).