6. The matrix \(\mathbf { M }\) is given by
$$\mathbf { M } = \left( \begin{array} { r r r }
1 & k & 0
2 & - 2 & 1
- 4 & 1 & - 1
\end{array} \right) , k \in \mathbb { R } , k \neq \frac { 1 } { 2 }$$
- Show that \(\operatorname { det } \mathbf { M } = 1 - 2 k\).
- 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
$$\left( \begin{array} { r r r }
1 & 0 & 0
2 & - 2 & 1
- 4 & 1 & - 1
\end{array} \right)$$
Given that \(l _ { 2 }\) has cartesian equation
$$\frac { x - 1 } { 5 } = \frac { y + 2 } { 2 } = \frac { z - 3 } { 1 }$$ - find a cartesian equation of the line \(l _ { 1 }\)