| Exam Board | Edexcel |
|---|---|
| Module | F3 (Further Pure Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Matrix properties verification |
| Difficulty | Moderate -0.8 This is a straightforward verification question testing basic matrix operations (transpose and multiplication) with symbolic entries. Part (a) is trivial recall, part (b) is routine 3×3 matrix multiplication, and part (c) requires no insight—students simply compute both sides and verify equality. The presence of parameters a and b adds minimal difficulty since they're treated algebraically throughout. This is easier than average A-level content, being a direct application of standard matrix properties without problem-solving. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\mathbf{A}^T = \begin{pmatrix} -1 & 2 & 1 \\ 3 & 0 & -2 \\ a & 1 & 1 \end{pmatrix}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\mathbf{AB} = \begin{pmatrix} a+7 & 2a-6 & ab+5 \\ 5 & 2 & b+8 \\ -3 & 6 & b-2 \end{pmatrix}\) | M1A1 | M1: Correct attempt at \(\mathbf{AB}\), min 5 entries correct; A1: Correct matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \((\mathbf{AB})^T = \begin{pmatrix} a+7 & 5 & -3 \\ 2a-6 & 2 & 6 \\ ab+5 & b+8 & b-2 \end{pmatrix}\) | B1 | Transposed matrix must be seen |
| \(\mathbf{B}^T\mathbf{A}^T = \begin{pmatrix} 2 & 3 & 1 \\ 0 & -2 & 2 \\ 4 & 3 & b \end{pmatrix} \begin{pmatrix} -1 & 2 & 1 \\ 3 & 0 & -2 \\ a & 1 & 1 \end{pmatrix}\) | Attempt \(\mathbf{B}^T\mathbf{A}^T\); must be in this order | |
| \(= \begin{pmatrix} a+7 & 5 & -3 \\ 2a-6 & 2 & 6 \\ ab+5 & b+8 & b-2 \end{pmatrix}\) | M1 | Must see matrices being multiplied; min 5 entries correct, follow through their errors in transposing \(\mathbf{A}\) and \(\mathbf{B}\) |
| \(\therefore (\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T\) | A1cso | Clearly shows \((\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T\) with conclusion and no errors; e.g. state \((\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T\) or connect through working by \(=\) signs; or QED, hence shown, \#, (list not exhaustive) |
| Part (c) Total: 3 marks | Question Total: 6 marks |
## Question 2: Matrices $\mathbf{A}$ and $\mathbf{B}$
### Part (a)
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\mathbf{A}^T = \begin{pmatrix} -1 & 2 & 1 \\ 3 & 0 & -2 \\ a & 1 & 1 \end{pmatrix}$ | B1 | |
**Part (a) Total: 1 mark**
### Part (b)
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\mathbf{AB} = \begin{pmatrix} a+7 & 2a-6 & ab+5 \\ 5 & 2 & b+8 \\ -3 & 6 & b-2 \end{pmatrix}$ | M1A1 | M1: Correct attempt at $\mathbf{AB}$, min 5 entries correct; A1: Correct matrix |
**Part (b) Total: 2 marks**
### Part (c)
| Working/Answer | Mark | Guidance |
|---|---|---|
| $(\mathbf{AB})^T = \begin{pmatrix} a+7 & 5 & -3 \\ 2a-6 & 2 & 6 \\ ab+5 & b+8 & b-2 \end{pmatrix}$ | B1 | Transposed matrix must be seen |
| $\mathbf{B}^T\mathbf{A}^T = \begin{pmatrix} 2 & 3 & 1 \\ 0 & -2 & 2 \\ 4 & 3 & b \end{pmatrix} \begin{pmatrix} -1 & 2 & 1 \\ 3 & 0 & -2 \\ a & 1 & 1 \end{pmatrix}$ | | Attempt $\mathbf{B}^T\mathbf{A}^T$; must be in this order |
| $= \begin{pmatrix} a+7 & 5 & -3 \\ 2a-6 & 2 & 6 \\ ab+5 & b+8 & b-2 \end{pmatrix}$ | M1 | Must see matrices being multiplied; min 5 entries correct, follow through their errors in transposing $\mathbf{A}$ and $\mathbf{B}$ |
| $\therefore (\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T$ | A1cso | Clearly shows $(\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T$ with conclusion and no errors; e.g. state $(\mathbf{AB})^T = \mathbf{B}^T\mathbf{A}^T$ or connect through working by $=$ signs; or QED, hence shown, \#, (list not exhaustive) |
**Part (c) Total: 3 marks | Question Total: 6 marks**
---2.
$$\mathbf { A } = \left( \begin{array} { r r r }
- 1 & 3 & a \\
2 & 0 & 1 \\
1 & - 2 & 1
\end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r r }
2 & 0 & 4 \\
3 & - 2 & 3 \\
1 & 2 & b
\end{array} \right)$$
where $a$ and $b$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Write down $\mathbf { A } ^ { \mathrm { T } }$ in terms of $a$.
\item Calculate $\mathbf { A B }$, giving your answer in terms of $a$ and $b$.
\item Hence show that
$$( \mathbf { A B } ) ^ { \mathrm { T } } = \mathbf { B } ^ { \mathrm { T } } \mathbf { A } ^ { \mathrm { T } }$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel F3 2017 Q2 [6]}}