| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | December |
| Marks | 5 |
| Topic | Matrices |
| Type | Solving matrix equations for unknown matrix |
| Difficulty | Standard +0.3 Part (a) requires solving BA² = A for B, which involves computing A², then finding B = A(A²)⁻¹ = A⁻¹—a straightforward matrix algebra exercise. Part (b) asks to verify that (I-C)(I+C) = I by computing C² and using the difference of squares pattern—routine algebraic manipulation. Both parts are standard Further Maths matrix questions requiring only direct application of techniques with no novel insight, placing them slightly above average A-level difficulty due to the Further Maths context. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03n Inverse 2x2 matrix4.03o Inverse 3x3 matrix |
\begin{enumerate}[label=\alph*)]
\item The $2 \times 2$ matrix A is given by
$$\mathbf{A} = \begin{pmatrix} 7 & 3 \\ 2 & 1 \end{pmatrix}.$$
The $2 \times 2$ matrix B satisfies
$$\mathbf{BA}^2 = \mathbf{A}.$$
Find the matrix B. [3]
\item The $2 \times 2$ matrix C is given by
$$\mathbf{C} = \begin{pmatrix} -2 & 4 \\ -1 & 2 \end{pmatrix}.$$
By considering $\mathbf{C}^2$, show that the matrices $\mathbf{I} - \mathbf{C}$ and $\mathbf{I} + \mathbf{C}$ are inverse to each other. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q8 [5]}}