5. (i)
$$\mathbf { A } = \left( \begin{array} { l l }
p & 2
3 & p
\end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r }
- 5 & 4
6 & - 5
\end{array} \right)$$
where \(p\) is a constant.
- Find, in terms of \(p\), the matrix \(\mathbf { A B }\)
Given that
$$\mathbf { A B } + 2 \mathbf { A } = k \mathbf { I }$$
where \(k\) is a constant and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix,
- find the value of \(p\) and the value of \(k\).
(ii)
$$\mathbf { M } = \left( \begin{array} { r r }
a & - 9
1 & 2
\end{array} \right) , \text { where } a \text { is a real constant }$$
Triangle \(T\) has an area of 15 square units.
Triangle \(T\) is transformed to the triangle \(T ^ { \prime }\) by the transformation represented by the matrix M.
Given that the area of triangle \(T ^ { \prime }\) is 270 square units, find the possible values of \(a\).