A2 October 2020 Paper 1 Q2
2
(a) The matrices \(\mathbf{M} = \begin{pmatrix} 0 & 1 & a \\ 1 & b & 0 \end{pmatrix}\) and \(\mathbf{N} = \begin{pmatrix} b & -5 \\ -1 & c \\ -1 & 1 \end{pmatrix}\) are such that \(\mathbf{MN} = \mathbf{I}\).
Find \(a\), \(b\) and \(c\). [5]
Find \(a\), \(b\) and \(c\). [5]
(b) State with a reason whether or not \(\mathbf{N}\) is the inverse of \(\mathbf{M}\). [1]
| Scheme | Marks | AO |
|---|---|---|
| \(\begin{pmatrix} 0 & 1 & a \\ 1 & b & 0 \end{pmatrix}\begin{pmatrix} b & -5 \\ -1 & c \\ -1 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) | B1 | 1.1a |
| \(-1 - a = 1 \Rightarrow a = -2\) \(c + a = 0 \Rightarrow c = 2\) \(-5 + bc = 1 \Rightarrow b = 3\) | M1 A1 A1ft A1 | 1.1 1.1 1.1 1.1 |
| [5] |
Notes
B1: \(\mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\). Soi
M1: matrix multiplication
A1: 3 correct equations
A1ft: their value of \(a\) from their equations
A1: all 3 values correct
| Scheme | Marks | AO |
|---|---|---|
| \(\mathbf{M}\) is not a square matrix and so has no inverse | B1 | 2.4 |
| [1] |
Notes
B1: \(\mathbf{MN} \neq \mathbf{NM}\)
different orders