| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2021 |
| Session | March |
| Marks | 8 |
| Topic | Proof by induction |
| Type | Prove matrix power formula |
| Difficulty | Standard +0.3 This is a standard matrix induction proof requiring verification of the base case and inductive step with matrix multiplication. While it involves Further Maths content (matrices), the induction structure is routine and the matrix multiplication is straightforward due to the upper triangular form with simple entries. The algebraic manipulation needed is minimal compared to typical proof questions. |
| Spec | 4.01a Mathematical induction: construct proofs4.03b Matrix operations: addition, multiplication, scalar |
$$\text{M} = \begin{pmatrix} 2 & 3 \\ 0 & 1 \end{pmatrix}$$ [2]
Prove by induction that $\text{M}^n = \begin{pmatrix} 2^n & 3(2^n - 1) \\ 0 & 1 \end{pmatrix}$, for all positive integers $n$. [6]
\hfill \mbox{\textit{SPS SPS FM 2021 Q6 [8]}}