13 Matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c c } k & 1 & - 5
2 & 3 & - 3
- 1 & 2 & 2 \end{array} \right)\), where \(k\) is a constant.
- Show that \(\operatorname { det } \mathbf { M } = 12 ( k - 3 )\).
- Find a solution of the following simultaneous equations for which \(x \neq z\).
$$\begin{aligned}
4 x ^ { 2 } + y ^ { 2 } - 5 z ^ { 2 } & = 6
2 x ^ { 2 } + 3 y ^ { 2 } - 3 z ^ { 2 } & = 6
- x ^ { 2 } + 2 y ^ { 2 } + 2 z ^ { 2 } & = - 6
\end{aligned}$$ - (A) Verify that the point ( \(2,0,1\) ) lies on each of the following three planes.
$$\begin{aligned}
3 x + y - 5 z & = 1
2 x + 3 y - 3 z & = 1
- x + 2 y + 2 z & = 0
\end{aligned}$$
(B) Describe how the three planes in part (iii) (A) are arranged in 3-D space. Give reasons for your answer. - Find the values of \(k\) for which the transformation represented by \(\mathbf { M }\) has a volume scale factor of 6 .