7. \(\quad \mathbf { A } = \left( \begin{array} { c c c } 2 & \mathrm { k } & 0
1 & 1 & 0
0 & - 2 & 1 \end{array} \right)\), where k is a constant.
Given that \(\left( \begin{array} { c } 9
3
- 2 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\),
- show that \(\mathrm { k } = 6\),
- find the eigenvalues of \(\mathbf { A }\).
A transformation \(\mathrm { T } : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { A }\).
The point P has coordinates \(( \mathrm { t } - 2 , \mathrm { t } , 2 \mathrm { t } )\) where t is a parameter. - Show that, for any value of \(t\), the transformation \(T\) maps \(P\) onto a point on the line with equation \(x - 4 y - 4 = 0\)
(5)