Problem

The matrix $P$ is given by

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

(a) State the eigenvalues of $P$.

(b) Use the characteristic equation of $P$ to find $P^{-1}$.

The $3\times3$ matrix $A$ has distinct non-zero eigenvalues $a$, $\frac{1}{2}$, $2$ with corresponding eigenvectors

$$\begin{pmatrix}1\\0\\0\end{pmatrix},\qquad \begin{pmatrix}-1\\2\\0\end{pmatrix},\qquad \begin{pmatrix}1\\1\\-1\end{pmatrix}$$

respectively.

(c) Find $A^{-1}$ in terms of $a$.