- The matrix \(\mathbf { M }\) is given by
$$\mathbf { M } = \left( \begin{array} { r r r }
2 & 1 & 0
1 & 2 & 0
- 1 & 0 & 4
\end{array} \right)$$
- Show that 4 is an eigenvalue of \(\mathbf { M }\), and find the other two eigenvalues.
- For the eigenvalue 4, find a corresponding eigenvector.
The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\mathbf { M }\).
The equation of \(l _ { 1 }\) is \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { b } = 0\), where \(\mathbf { a } = 3 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\).
- Find a vector equation for the line \(l _ { 2 }\).