To find the eigenvalues and eigenvectors of the matrix
\[
A = \begin{bmatrix}
4 & 0 & 1 \\
-2 & 1 & 0 \\
-2 & 0 & 1
\end{bmatrix},
\]
we begin by calculating the eigenvalues. The eigenvalues can be found by solving the characteristic polynomial, which is given by
\[
\det(A - \lambda I = 0),
\]
where \( \lambda \) is the eigenvalue and \( I \) is the identity matrix. Thus, we first compute \( A - \lambda I \):
\[
A - \lambda I =