$$\mathbf { A } = \left( \begin{array} { r r }
5 k & 3 k - 1
- 3 & k + 1
\end{array} \right) , \text { where } k \text { is a real constant. }$$
Given that \(\mathbf { A }\) is a singular matrix, find the possible values of \(k\).
(ii)
$$\mathbf { B } = \left( \begin{array} { l l }
10 & 5
- 3 & 3
\end{array} \right)$$
A triangle \(T\) is transformed onto a triangle \(T ^ { \prime }\) by the transformation represented by the matrix \(\mathbf { B }\).
The vertices of triangle \(T ^ { \prime }\) have coordinates \(( 0,0 ) , ( - 20,6 )\) and \(( 10 c , 6 c )\), where \(c\) is a positive constant.
The area of triangle \(T ^ { \prime }\) is 135 square units.
- Find the matrix \(\mathbf { B } ^ { - 1 }\)
- Find the coordinates of the vertices of the triangle \(T\), in terms of \(c\) where necessary.
- Find the value of \(c\).