12 The matrix \(\mathbf { A } = \left[ \begin{array} { c c c } 1 & 5 & 3
4 & - 2 & p
8 & 5 & - 11 \end{array} \right]\), where \(p\) is a constant.
12
- Given that \(\mathbf { A }\) is a non-singular matrix, find \(\mathbf { A } ^ { - 1 }\) in terms of \(p\).
State any restrictions on the value of \(p\).
12 - The equations below represent three planes.
$$\begin{aligned}
x + 5 y + 3 z & = 5
4 x - 2 y + p z & = 24
8 x + 5 y - 11 z & = - 30
\end{aligned}$$
12 - Find, in terms of \(p\), the coordinates of the point of intersection of the three planes.
[0pt]
[4 marks]
12
- (ii) In the case where \(p = 2\), show that the planes are mutually perpendicular.