AS June 2022 Paper 1 Q6
6 The matrix \(\mathbf{A}\) is given by
\[\mathbf{A} = \begin{bmatrix} 5 & 2 \\ -3 & 4 \end{bmatrix}\](a) Find \(\det\mathbf{A}\) [1 mark]
(b) Find \(\mathbf{A}^{-1}\) [1 mark]
(c) Given that \(\mathbf{AB} = \begin{bmatrix} 9 & 6 \\ 5 & 12 \end{bmatrix}\) and \(\mathbf{M} = 2\mathbf{A} + \mathbf{B}\) find the matrix \(\mathbf{M}\) [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Obtains the correct determinant. | B1 | 1.1b |
| (1) |
Typical solution
\[\begin{aligned}\det\mathbf{A} &= 5 \times 4 - (-3) \times 2 \\ &= 26\end{aligned}\]| Scheme | Marks | AO |
|---|---|---|
| Obtains the correct inverse matrix ACF FT their determinant | B1F | 1.1b |
| (1) |
Typical solution
\[\mathbf{A}^{-1} = \frac{1}{26}\begin{bmatrix} 4 & -2 \\ 3 & 5 \end{bmatrix}\]| Scheme | Marks | AO |
|---|---|---|
| Selects a method to find either matrix \(\mathbf{B}\) or matrix \(\mathbf{AM}\) e.g. calculates \(\mathbf{A}^{-1}\mathbf{AB}\) with their \(\mathbf{A}^{-1}\) or calculates \(2\mathbf{A}^2 + \mathbf{AB}\) or writes four simultaneous equations in \(w, x, y, z\) where \(\begin{bmatrix} 9 & 6 \\ 5 & 12 \end{bmatrix} = \begin{bmatrix} 5 & 2 \\ -3 & 4 \end{bmatrix}\begin{bmatrix} w & x \\ y & z \end{bmatrix}\) | M1 | 3.1a |
| Obtains a correct matrix for \(\mathbf{B}\) or \(\mathbf{AM}\) i.e. \(\begin{bmatrix} 1 & 0 \\ 2 & 3 \end{bmatrix}\) or \(\begin{bmatrix} 47 & 42 \\ -49 & 32 \end{bmatrix}\) PI by a correct matrix \(\mathbf{M}\) FT their \(\mathbf{A}^{-1}\) | A1F | 1.1b |
| Obtains matrix \(\mathbf{M}\) FT their \(\mathbf{A}^{-1}\) | A1F | 2.2a |
| (3) | ||
| (5 marks) |