Vector equation of a plane

5. $$\mathbf { M } = \left( \begin{array} { r r r } 4 & - 5 & 0 \\ k & 2 & 0 \\ - 3 & - 5 & k \end{array} \right) \text {, where } k \text { is a real constant, } k \neq 0 , k \neq - \frac { 8 } { 5 }$$

1. Find, in terms of \(k\), the inverse of the matrix \(\mathbf { M }\). A transformation \(T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix $$\left( \begin{array} { r r r } 4 & - 5 & 0 \\ - 1 & 2 & 0 \\ - 3 & - 5 & - 1 \end{array} \right)$$ The transformation \(T\) maps the plane \(\Pi _ { 1 }\) onto the plane \(\Pi _ { 2 }\)
   Given that the plane \(\Pi _ { 2 }\) has equation \(2 x - z = 4\)
2. find a cartesian equation of the plane \(\Pi _ { 1 }\)

1. The matrix \(\mathbf { M }\) is given by

$$\mathbf { M } = \left( \begin{array} { r r r } k & - 1 & 1 \\ 1 & 0 & - 1 \\ 3 & - 2 & 1 \end{array} \right) , \quad k \neq 1$$

1. Show that \(\operatorname { det } \mathbf { M } = 2 - 2 k\).
2. 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 \(\mathbf { b } = 4 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }\).
3. Find a vector equation for the line \(l _ { 1 }\).

1. The plane \(\Pi _ { 1 }\) has vector equation

$$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) + t \left( \begin{array} { r } 1 \\ 2 \\ - 2 \end{array} \right) ,$$ where \(s\) and \(t\) are real parameters. The plane \(\Pi _ { 1 }\) is transformed to the plane \(\Pi _ { 2 }\) by the transformation represented by the matrix \(\mathbf { T }\), where $$\mathbf { T } = \left( \begin{array} { r r r } 2 & 0 & 3 \\ 0 & 2 & - 1 \\ 0 & 1 & 2 \end{array} \right)$$ Find an equation of the plane \(\Pi _ { 2 }\) in the form r.n=p