A2 June 2020 Paper 2 Q4
4 The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are defined as follows:
\[\mathbf{A} = \begin{bmatrix} x + 1 & 2 \\ x + 2 & -3 \end{bmatrix} \qquad \mathbf{B} = \begin{bmatrix} x - 4 & x - 2 \\ 0 & -2 \end{bmatrix}\]Show that there is a value of \(x\) for which \(\mathbf{AB} = k\mathbf{I}\), where \(\mathbf{I}\) is the \(2 \times 2\) identity matrix and \(k\) is an integer to be found. [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Multiplies matrices \(\mathbf{A}\) and \(\mathbf{B}\) to form the product \(\mathbf{AB}\) with at least one element of the product correct. Condone \(\mathbf{BA}\) | M1 | 1.1a |
| Forms the correct product \(\mathbf{AB}\) (may be unsimplified) | A1 | 1.1b |
| Deduces from correct product that \(\mathbf{AB} = 6\mathbf{I}\), when \(x = -2\) | R1 | 2.2a |
| (3 marks) |