Problem

The matrix $M$ is given by

$$M = \begin{pmatrix} 3 & 1 & p \\ 1 & 1 & 2 \\ -1 & p & 2 \end{pmatrix}$$

where $p$ is a real constant.

(a) Find the exact values of $p$ for which $M$ has no inverse.

Given that $M$ does have an inverse,

(b) find $M^{-1}$ in terms of $p$.