- The matrix \(\mathbf { M }\) is given by
$$\mathbf { M } = \left( \begin{array} { r r r }
k & - 1 & 1
1 & 0 & - 1
3 & - 2 & 1
\end{array} \right) , \quad k \neq 1$$
- Show that \(\operatorname { det } \mathbf { M } = 2 - 2 k\).
- Find \(\mathbf { M } ^ { - 1 }\), in terms of \(k\).
The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\left( \begin{array} { r r r } 2 & - 1 & 1
1 & 0 & - 1
3 & - 2 & 1 \end{array} \right)\).
The equation of \(l _ { 2 }\) is \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { b } = 0\), where \(\mathbf { a } = 4 \mathbf { i } + \mathbf { j } + 7 \mathbf { k }\) and \(\mathbf { b } = 4 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }\). - Find a vector equation for the line \(l _ { 1 }\).