AS June 2021 Paper 1 Q10
10 Matrix \(\mathbf{A}\) is given by
\[\mathbf{A} = \begin{bmatrix} 3 & \mathrm{i} - 1 \\ \mathrm{i} & 2 \end{bmatrix}\](a) Show that \(\det \mathbf{A} = a + \mathrm{i}\) where \(a\) is an integer to be determined. [2 marks]
(b) Matrix \(\mathbf{B}\) is given by\[\mathbf{B} = \begin{bmatrix} 14 - 2\mathrm{i} & b \\ c & d \end{bmatrix} \quad \text{and} \quad \mathbf{AB} = p\mathbf{I}\]
where \(b, c, d \in \mathbb{C}\) and \(p \in \mathbb{N}\)
Find \(b\), \(c\), \(d\) and \(p\) [6 marks]
| Scheme | Marks | AO |
|---|---|---|
| Writes a correct unsimplified expression for \(\det \mathbf{A}\) | M1 | 1.1a |
| Completes a fully correct proof to reach the required result. Must have \(a = 7\) | R1 | 2.1 |
| (2) |
Typical solution
\[\begin{aligned}3 \times 2 - \mathrm{i}(\mathrm{i} - 1) &= 6 - \mathrm{i}^2 + \mathrm{i} \\ &= 7 + \mathrm{i}\end{aligned}\]| Scheme | Marks | AO |
|---|---|---|
| Recognises that \(\mathbf{B}\) is equal to a multiple of \(\mathbf{A}^{-1}\) Or multiplies \(\mathbf{A}\) and \(\mathbf{B}\) to find at least one correct unsimplified element of \(\mathbf{AB}\) | M1 | 3.1a |
| Sets up at least one correct non-matrix equation in one or two unknowns by equating a pair of corresponding elements. | M1 | 1.1a |
| Sets up another correct non-matrix equation in one unknown only. | M1 | 1.1a |
| Obtains at least one correct value of \(p\), \(b\), \(c\) or \(d\). Could be seen as an element of a matrix, e.g. \(\tfrac{1}{50}\begin{bmatrix} 14 - 2\mathrm{i} & * \\ * & * \end{bmatrix}\) or \(k\begin{bmatrix} 14 - 2\mathrm{i} & 6 - 8\mathrm{i} \\ * & * \end{bmatrix}\) or \(k\begin{bmatrix} 14 - 2\mathrm{i} & * \\ -1 - 7\mathrm{i} & * \end{bmatrix}\) or \(k\begin{bmatrix} 14 - 2\mathrm{i} & * \\ * & 21 - 3\mathrm{i} \end{bmatrix}\) | A1 | 1.1b |
| Obtains at least two correct values of \(p\), \(b\), \(c\) or \(d\). Could be seen as an element of a matrix, e.g. \(\tfrac{1}{50}\begin{bmatrix} 14 - 2\mathrm{i} & * \\ * & * \end{bmatrix}\) or \(k\begin{bmatrix} 14 - 2\mathrm{i} & 6 - 8\mathrm{i} \\ * & * \end{bmatrix}\) or \(k\begin{bmatrix} 14 - 2\mathrm{i} & * \\ -1 - 7\mathrm{i} & * \end{bmatrix}\) or \(k\begin{bmatrix} 14 - 2\mathrm{i} & * \\ * & 21 - 3\mathrm{i} \end{bmatrix}\) | A1 | 1.1b |
| Obtains all four correct values of \(p\), \(b\), \(c\) and \(d\). Accept \(p = 50\) and \(\mathbf{B} = \begin{bmatrix} 14 - 2\mathrm{i} & 6 - 8\mathrm{i} \\ -1 - 7\mathrm{i} & 21 - 3\mathrm{i} \end{bmatrix}\) | A1 | 1.1b |
| (6) | ||
| (8 marks) |