Problem

The matrix $\mathbf{P}$ has non-zero eigenvalues and is given by

$$\mathbf{P} = \begin{pmatrix} a & 1 & 1\\ 0 & 2a & -1\\ 0 & 0 & -3a \end{pmatrix}.$$

(a) State, in terms of $a$, the eigenvalues of $\mathbf{P}$.

(b) (i) Find $\mathbf{P}^2$ in terms of $a$.

(ii) Use the characteristic equation of $\mathbf{P}$ to find $\mathbf{P}^{-1}$ in terms of $a$.

The $3 \times 3$ matrix $\mathbf{A}$ has eigenvalues $1$, $2$, $3$ with corresponding eigenvectors

$$\begin{pmatrix} a\\ 0\\ 0 \end{pmatrix}, \quad \begin{pmatrix} 1\\ 2a\\ 0 \end{pmatrix}, \quad \begin{pmatrix} 1\\ -1\\ -3a \end{pmatrix},$$

respectively.

(c) Find $\mathbf{A}$ in terms of $a$.