| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2022 |
| Session | February |
| Marks | 8 |
| Topic | Proof by induction |
| Type | Prove matrix power formula |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question on matrix powers and proof by induction. Part (i) involves simple diagonal matrix multiplication (trivial since diagonal matrices just require raising diagonal entries to powers), part (ii) is pattern recognition, and part (iii) is a standard induction proof with no complications since diagonal matrices multiply easily. While it's a multi-part question worth 8 marks, it requires only routine techniques with no problem-solving insight, making it slightly easier than average overall. |
| Spec | 4.01a Mathematical induction: construct proofs4.03b Matrix operations: addition, multiplication, scalar |
The matrix $\mathbf{A}$ is given by $\mathbf{A} = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$.
\begin{enumerate}[label=(\roman*)]
\item Find $\mathbf{A}^2$ and $\mathbf{A}^3$. [3]
\item Hence suggest a suitable form for the matrix $\mathbf{A}^n$. [1]
\item Use induction to prove that your answer to part (ii) is correct. [4]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2022 Q7 [8]}}