9 You are given that matrix \(\mathbf { M } = \left( \begin{array} { l l } - 3 & 8
- 2 & 5 \end{array} \right)\).
- Prove that, for all positive integers \(n , \mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 - 4 n & 8 n
- 2 n & 1 + 4 n \end{array} \right)\). - Determine the equation of the line of invariant points of the transformation represented by the matrix \(\mathbf { M }\).
It is claimed that the answer to part (ii) is also a line of invariant points of the transformation represented by the matrix \(\mathbf { M } ^ { n }\), for any positive integer \(n\).
- Explain geometrically why this claim is true.
- Verify algebraically that this claim is true.
\section*{END OF QUESTION PAPER}
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