Matrix powers and patterns

5. \(\mathbf { R } = \left( \begin{array} { l l } a & 2 \\ a & b \end{array} \right)\), where \(a\) and \(b\) are constants and \(a > 0\).

1. Find \(\mathbf { R } ^ { 2 }\) in terms of \(a\) and \(b\). Given that \(\mathbf { R } ^ { 2 }\) represents an enlargement with centre ( 0,0 ) and scale factor 15 ,
2. find the value of \(a\) and the value of \(b\).

6 You are given that \(\mathbf { M } = \left( \begin{array} { l l } 4 & - 9 \\ 1 & - 2 \end{array} \right)\).

1. Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 + 3 n & - 9 n \\ n & 1 - 3 n \end{array} \right)\) for all positive integers \(n\).
2. A student thinks that this formula, when \(n = 0\) and \(n = - 1\), gives the identity matrix and the inverse matrix \(\mathbf { M } ^ { - 1 }\) respectively. Determine whether the student is correct.

11 Prove by induction that, for all integers \(n \geq 1\), $$\left( \mathbf { A B A } ^ { - 1 } \right) ^ { n } = \mathbf { A B } ^ { n } \mathbf { A } ^ { - 1 }$$ where \(\mathbf { A }\) and \(\mathbf { B }\) are square matrices of equal dimensions, and \(\mathbf { A }\) is non-singular.

8 In this question you must show detailed reasoning. \(\mathbf { M }\) is the matrix \(\left( \begin{array} { l l } 1 & 6 \\ 0 & 2 \end{array} \right)\).
Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & 3 \left( 2 ^ { n + 1 } - 2 \right) \\ 0 & 2 ^ { n } \end{array} \right)\), for any positive integer \(n\). \section*{END OF QUESTION PAPER}