| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2021 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Matrices |
| Type | Properties of matrix operations |
| Difficulty | Moderate -0.3 This question tests understanding of matrix multiplication being non-commutative, a fundamental property taught early in Further Maths. Part (a) requires spotting that BA^{-1} ≠ A^{-1}B (line 2 is incorrect), and part (b) requires stating the correct result (AB)^{-1} = B^{-1}A^{-1} and fixing the proof. While it requires conceptual understanding rather than just computation, this is a standard textbook result that students are explicitly taught, making it slightly easier than average for a Further Maths question. |
| Spec | 4.03p Inverse properties: (AB)^(-1) = B^(-1)*A^(-1) |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (a) | −1≠A −1 |
| Answer | Marks |
|---|---|
| as matrix multiplication is not commutative | M1 |
| Answer | Marks |
|---|---|
| [2] | 2.3 |
| Answer | Marks | Guidance |
|---|---|---|
| 4 | (b) | Correct inverse of AB is B−1A−1 |
| Answer | Marks |
|---|---|
| [=AIA −1]=AA −1=I | B1 |
| Answer | Marks |
|---|---|
| [3] | 2.1 |
| Answer | Marks |
|---|---|
| 2.2a | BB −1 =I soi [need not show I] |
| Answer | Marks |
|---|---|
| …AA | SCB1 for a correct |
Question 4:
4 | (a) | −1≠A −1
BA B
as matrix multiplication is not commutative | M1
A1
[2] | 2.3
2.4
4 | (b) | Correct inverse of AB is B−1A−1
−1 −1)=A(BB −1)A −1
(AB)(B A
[=AIA −1]=AA −1=I | B1
M1
A1
[3] | 2.1
2.1
2.2a | BB −1 =I soi [need not show I]
−1=I
…AA | SCB1 for a correct
proof but not using
Anika’s working4 Anika thinks that, for two square matrices $\mathbf { A }$ and $\mathbf { B }$, the inverse of $\mathbf { A B }$ is $\mathbf { A } ^ { - 1 } \mathbf { B } ^ { - 1 }$. Her attempted proof of this is as follows.
$$\begin{aligned}
( \mathbf { A B } ) \left( \mathbf { A } ^ { - 1 } \mathbf { B } ^ { - 1 } \right) & = \mathbf { A } \left( \mathbf { B A } ^ { - 1 } \right) \mathbf { B } ^ { - 1 } \\
& = \mathbf { A } \left( \mathbf { A } ^ { - 1 } \mathbf { B } \right) \mathbf { B } ^ { - 1 } \\
& = \left( \mathbf { A } \mathbf { A } ^ { - 1 } \right) \left( \mathbf { B B } ^ { - 1 } \right) \\
& = \mathbf { I } \times \mathbf { I } \\
& = \mathbf { I } \\
\text { Hence } ( \mathbf { A B } ) ^ { - 1 } & = \mathbf { A } ^ { - 1 } \mathbf { B } ^ { - 1 }
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Explain the error in Anika's working.
\item State the correct inverse of the matrix $\mathbf { A B }$ and amend Anika's working to prove this.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2021 Q4 [5]}}