7. (i)
$$\mathbf { A } = \left( \begin{array} { r r }
6 & k
- 3 & - 4
\end{array} \right) , \text { where } k \text { is a real constant, } k \neq 8$$
Find, in terms of \(k\),
- \(\mathbf { A } ^ { - 1 }\)
- \(\mathbf { A } ^ { 2 }\)
Given that \(\mathbf { A } ^ { 2 } + 3 \mathbf { A } ^ { - 1 } = \left( \begin{array} { r r } 5 & 9
- 3 & - 5 \end{array} \right)\) - find the value of \(k\).
(ii)
$$\mathbf { M } = \left( \begin{array} { c c }
- \frac { 1 } { 2 } & - \sqrt { 3 }
\frac { \sqrt { 3 } } { 2 } & - 1
\end{array} \right)$$
The matrix \(\mathbf { M }\) represents a one way stretch, parallel to the \(y\)-axis, scale factor \(p\), where \(p > 0\), followed by a rotation anticlockwise through an angle \(\theta\) about \(( 0,0 )\). - Find the value of \(p\).
- Find the value of \(\theta\).