6. \(\mathbf { M } = \left( \begin{array} { c c c } 1 & 0 & 3
0 & - 2 & 1
k & 0 & 1 \end{array} \right)\), where \(k\) is a constant.
Given that \(\left( \begin{array} { l } 6
1
6 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\),
- find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { l } 6
1
6 \end{array} \right)\), - show that \(k = 3\),
- show that \(\mathbf { M }\) has exactly two eigenvalues.
A transformation \(T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by \(\mathbf { M }\).
The transformation \(T\) maps the line \(l _ { 1 }\), with cartesian equations \(\frac { x - 2 } { 1 } = \frac { y } { - 3 } = \frac { z + 1 } { 4 }\), onto the line \(l _ { 2 }\). - Taking \(k = 3\), find cartesian equations of \(l _ { 2 }\).