The matrix $A$ is given as:

$A = \begin{bmatrix} 4 & 0 & -1 \\ 0 & 1 & -4 \\ 0 & 0 & 0 \end{bmatrix}$

To find the eigenvalues, we solve the characteristic equation:

$\text{det}(A - \lambda I) = 0$

where $I$ is the identity matrix and $\lambda$ is the eigenvalue.

Step 1: Set up the characteristic matrix

$A - \lambda I = \begin{bmatrix} 4 - \lambda & 0 & -1 \\ 0 & 1 - \lambda & -4 \\ 0 & 0 & -\lambda \end{bmatrix}$

Step 2: Calculate the determinant

The determinant is:

$\text{det}(A - \lambda I) = (4 - \lambda)\begin{vmatrix} 1 - \lambda & -4 \\ 0 & -\lambda \end{vmatrix} = (4 - \lambda)((1 - \lambda)(-\lambda) - 0)$