| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Topic | Proof by induction |
| Type | Prove matrix power formula |
| Difficulty | Challenging +1.2 This is a proof by induction on matrices with complex entries. While it requires knowledge of matrix multiplication and complex number arithmetic, the structure is entirely standard: verify base case n=1, assume for n=k, prove for n=k+1. The matrix multiplication is straightforward (2×2 with simple patterns), and no insight beyond following the induction template is needed. Slightly above average due to complex entries and being Further Maths content, but remains a routine textbook exercise. |
| Spec | 4.01a Mathematical induction: construct proofs4.03b Matrix operations: addition, multiplication, scalar |
Prove that for all $n \in \mathbb{N}$
$$\begin{pmatrix} 3 & 4i \\ i & -1 \end{pmatrix}^n = \begin{pmatrix} 2n+1 & 4ni \\ ni & 1-2n \end{pmatrix}$$ [6]
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q7 [6]}}