| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2021 |
| Session | April |
| Marks | 6 |
| Topic | Proof by induction |
| Type | Prove matrix power formula |
| Difficulty | Standard +0.3 This is a standard matrix induction proof with straightforward matrix multiplication. The base case is trivial, and the inductive step requires only routine 2×2 matrix multiplication and simple algebraic manipulation of the formula 3(2^(k+1) - 1) = 3(2·2^k - 1). While it's a Further Maths question, it's a textbook example of proof by induction with matrices, requiring no novel insight or complex reasoning—slightly easier than average overall. |
| Spec | 4.01a Mathematical induction: construct proofs4.03b Matrix operations: addition, multiplication, scalar |
$$\mathbf{M} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}.$$
Prove by induction that $\mathbf{M}^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}$, for all positive integers $n$. [6]
\hfill \mbox{\textit{SPS SPS FM 2021 Q7 [6]}}