A2 June 2023 Paper 2 Q1
1
(a) The matrix \(\mathbf{P}\) is given by \(\mathbf{P} = \begin{pmatrix} 1 & 0 & -2 & 2 \\ 4 & 2 & -2 & 3 \end{pmatrix}\).
(i) Write down the dimensions of \(\mathbf{P}\). [1]
(ii) Write down the transpose of \(\mathbf{P}\). [1]
(b) The matrices \(\mathbf{Q}\), \(\mathbf{R}\) and \(\mathbf{S}\) are given by \(\mathbf{Q} = \begin{pmatrix} 1 & 2 \end{pmatrix}\), \(\mathbf{R} = \begin{pmatrix} 3 & -4 \\ 2 & 3 \end{pmatrix}\) and \(\mathbf{S} = \begin{pmatrix} 3 & -2 \end{pmatrix}\).
Write down the sum of the two of these matrices which are conformable for addition. [1]
(c) The dimensions of matrix \(\mathbf{A}\) are 4 by 5. The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are conformable for multiplication so that the matrix \(\mathbf{C} = \mathbf{BA}\) can be formed. The matrix \(\mathbf{C}\) has 6 rows.
(i) Write down the number of columns that \(\mathbf{C}\) has. [1]
(ii) Write down the dimensions of \(\mathbf{B}\). [1]
(iii) Explain whether the matrix \(\mathbf{AB}\) can be formed. [1]
(d) Find the value of \(c\) for which \(\begin{pmatrix} -2 & 3 \\ 6 & 10 \end{pmatrix}\begin{pmatrix} c & 5 \\ 10 & 13 \end{pmatrix} = \begin{pmatrix} c & 5 \\ 10 & 13 \end{pmatrix}\begin{pmatrix} -2 & 3 \\ 6 & 10 \end{pmatrix}\). [2]
| Scheme | Marks | AO |
|---|---|---|
| (i) 2 by 4 or \(2 \times 4\) | B1 | 1.2 |
| [1] | ||
| (ii) \(\left(\mathbf{P}^{\mathrm{T}} =\right)\begin{pmatrix} 1 & 4 \\ 0 & 2 \\ -2 & -2 \\ 2 & 3 \end{pmatrix}\) | B1 | 1.2 |
| [1] |
Notes
(a)(ii)
B1: condone poor/omitted brackets just here
| Scheme | Marks | AO |
|---|---|---|
| \(\begin{pmatrix} 4 & 0 \end{pmatrix}\) | B1 | 2.5 |
| [1] |
Notes
B1: Do not allow \((4, 0)\)
| Scheme | Marks | AO |
|---|---|---|
| (i) 5 | B1 | 2.2a |
| [1] | ||
| (ii) 6 by 4 | B1 | 2.2a |
| [1] | ||
| (iii) No because the number of columns in \(\mathbf{A}\) (is 5 which) is not equal to the number of rows in matrix \(\mathbf{B}\) (which is 6) (and for the matrices to be conformable these have to be the same.) | B1 | 2.4 |
| [1] |
Notes
(c)(iii)
B1: Must include “number of” oe. If numbers used, must have a word to imply comparison (eg “while”, “but” rather than “and”)
Accept \((4 \times 5) \times (6 \times 4)\) and “\(5 \neq 6\)”; numbers given must be correct
| Scheme | Marks | AO |
|---|---|---|
| \(\begin{pmatrix} -2 & 3 \\ 6 & 10 \end{pmatrix}\begin{pmatrix} c & 5 \\ 10 & 13 \end{pmatrix} = \begin{pmatrix} 30 - 2c & 29 \\ 6c + 100 & 160 \end{pmatrix}\) and \(\begin{pmatrix} c & 5 \\ 10 & 13 \end{pmatrix}\begin{pmatrix} -2 & 3 \\ 6 & 10 \end{pmatrix} = \begin{pmatrix} 30 - 2c & 3c + 50 \\ 58 & 160 \end{pmatrix}\) | M1 | 1.1 |
| \(6c + 100 = 58\) or \(3c + 50 = 29 \Rightarrow c = -7\) | A1 | 1.1 |
| [2] |
Notes
M1: Attempt at multiplication in both directions sufficient to obtain one pair of equivalent entries in trailing diagonal.
Can be implied by correct linear equation
A1: ignore errors in unused elements