Standard +0.3 This is a standard matrix induction proof requiring verification of the base case (n=1) and inductive step. While it's a Further Maths topic, the structure is routine: multiply the matrices, simplify using the inductive hypothesis, and verify the result matches the required form. The matrix multiplication is straightforward with many zero entries, making calculations simpler than typical induction proofs.
6 Prove by mathematical induction that \(\left( \begin{array} { r l } 2 & 0 \\ - 1 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 0 \\ 1 - 2 ^ { n } & 1 \end{array} \right)\) for all positive integers \(n\).
Answer all the questions.
Section B (107 marks)
6 Prove by mathematical induction that $\left( \begin{array} { r l } 2 & 0 \\ - 1 & 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 0 \\ 1 - 2 ^ { n } & 1 \end{array} \right)$ for all positive integers $n$.
Answer all the questions.\\
Section B (107 marks)
\hfill \mbox{\textit{OCR MEI Further Pure Core 2022 Q6 [5]}}