6 A linear transformation \(T\) of the \(x - y\) plane has an associated matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { c c } \lambda & k
1 & \lambda - k \end{array} \right)\), and \(\lambda\)
and \(k\) are real constants. and \(k\) are real constants.

1. You are given that \(\operatorname { det } \mathbf { M } > 0\) for all values of \(\lambda\).
   1. Find the range of possible values of \(k\).
   2. What is the significance of the condition \(\operatorname { det } \mathbf { M } > 0\) for the transformation T? For the remainder of this question, take \(k = - 2\).
2. Determine whether there are any lines through the origin that are invariant lines for the transformation T.
3. The transformation T is applied to a triangle with area 3 units \({ } ^ { 2 }\). The area of the resulting image triangle is 15 units \({ } ^ { 2 }\).
   Find the possible values of \(\lambda\).