Standard +0.3 This is a standard proof by induction on matrix powers with a triangular matrix. The base case is straightforward matrix multiplication, and the inductive step requires multiplying M^k by M and simplifying the (1,2) entry using the formula 3(2^(k+1) - 2) + 6ยท2^k = 3(2^(k+2) - 2). While it requires careful algebraic manipulation, this is a routine FP1 technique with no novel insight needed, making it slightly easier than average.
4 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 \(n\).
4 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 $n$.
\hfill \mbox{\textit{OCR FP1 AS 2021 Q4 [6]}}