$$\mathbf { A } = \left( \begin{array} { c c }
2 & a \\
a - 4 & b
\end{array} \right)$$
where \(a\) and \(b\) are non-zero constants.
Given that the matrix \(\mathbf { A }\) is self-inverse,
- determine the value of \(b\) and the possible values for \(a\).
The matrix \(\mathbf { A }\) represents a linear transformation \(M\).
Using the smaller value of \(a\) from part (a), - show that the invariant points of the linear transformation \(M\) form a line, stating the equation of this line.
(ii)
$$\mathbf { P } = \left( \begin{array} { c c }
p & 2 p \\
- 1 & 3 p
\end{array} \right)$$
where \(p\) is a positive constant.
The matrix \(\mathbf { P }\) represents a linear transformation \(U\).
The triangle \(T\) has vertices at the points with coordinates ( 1,2 ), ( 3,2 ) and ( 2,5 ). The area of the image of \(T\) under the linear transformation \(U\) is 15 - Determine the value of \(p\).
The transformation \(V\) consists of a stretch scale factor 3 parallel to the \(x\)-axis with the \(y\)-axis invariant followed by a stretch scale factor - 2 parallel to the \(y\)-axis with the \(x\)-axis invariant. The transformation \(V\) is represented by the matrix \(\mathbf { Q }\).
- Write down the matrix \(\mathbf { Q }\).
Given that \(U\) followed by \(V\) is the transformation \(W\), which is represented by the matrix \(\mathbf { R }\), (c) find the matrix \(\mathbf { R }\).