3 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } 1 & 0
0 & k \end{array} \right) \left( \begin{array} { c c } 1 & 0
k & 1 \end{array} \right)\), where \(k\) is a constant and \(k \neq 0\) or 1 .

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied.
2. Write \(\mathbf { M } ^ { - 1 }\) as the product of two matrices, neither of which is \(\mathbf { I }\).
3. Show that the invariant points of the transformation represented by \(\mathbf { M }\) lie on the line \(\mathrm { y } = \frac { \mathrm { k } ^ { 2 } } { 1 - \mathrm { k } } \mathrm { x }\). [4]
4. The triangle \(A B C\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(D E F\). Find the value of \(k\) for which the area of triangle \(D E F\) is equal to the area of triangle \(A B C\).