Problem

$\mathbf{P} = \begin{pmatrix} p - 1 & p + 1 \\ -3 & p \end{pmatrix}$

where $p$ is a constant

(a) Determine $\det \mathbf{P}$ in simplest form in terms of $p$.

(b) Hence show that $\mathbf{P}$ is non-singular for all real values of $p$.

(c) Determine $\mathbf{P}^{-1}$ in terms of $p$.