The matrix \(\mathbf{A}\) is defined by
$$\mathbf{A} = \begin{pmatrix} 4 & 8 & 0 \\ 0 & \lambda & -2 \\ 4 & 0 & \lambda \end{pmatrix}.$$
- Show that there are two values of \(\lambda\) for which \(\mathbf{A}\) is singular. [4]
- Given that \(\lambda = 3\),
- determine the adjugate matrix of \(\mathbf{A}\),
- determine the inverse matrix \(\mathbf{A}^{-1}\). [5]