AS October 2020 Paper 1 Q6
6 The matrices \(\mathbf{M}\) and \(\mathbf{N}\) are \(\begin{pmatrix} \lambda & 2 \\ 2 & \lambda \end{pmatrix}\) and \(\begin{pmatrix} \mu & 1 \\ 1 & \mu \end{pmatrix}\) respectively, where \(\lambda\) and \(\mu\) are constants.
(a) Investigate whether \(\mathbf{M}\) and \(\mathbf{N}\) are commutative under multiplication. [2]
(b) You are now given that \(\mathbf{MN} = \mathbf{I}\).
(i) Write down a relationship between \(\det\mathbf{M}\) and \(\det\mathbf{N}\). [1]
(ii) Given that \(\lambda \gt 0\), find the exact values of \(\lambda\) and \(\mu\). [3]
(iii) Hence verify your answer to part (i). [2]
| Scheme | Marks | AO |
|---|---|---|
| \(\mathbf{MN} = \begin{pmatrix} \lambda\mu + 2 & \lambda + 2\mu \\ \lambda + 2\mu & \lambda\mu + 2 \end{pmatrix}\) | M1 | 2.5 |
| \(\mathbf{NM} = \begin{pmatrix} \lambda\mu + 2 & \lambda + 2\mu \\ \lambda + 2\mu & \lambda\mu + 2 \end{pmatrix}\) So \(\mathbf{M}\) and \(\mathbf{N}\) are commutative | A1cao | 2.2a |
| [2] |
Notes
M1: Attempt to show \(\mathbf{MN} = \mathbf{NM}\)
| Scheme | Marks | AO |
|---|---|---|
| (i) \(\det\mathbf{M} \times \det\mathbf{N} = 1\) | B1 | 2.2a |
| [1] | ||
| (ii) \(\lambda\mu + 2 = 1,\ \lambda + 2\mu = 0\) | B1 | 3.1a |
| \(\Rightarrow \lambda = -2\mu,\ -2\mu^2 + 2 = 1\) | M1 | 1.1 |
| \(\Rightarrow \mu = -1/\sqrt{2},\ \lambda = \sqrt{2}\) | A1 | 1.1 |
| [3] | ||
| (iii) \(\det\mathbf{M} = \lambda^2 - 4 = -2,\ \det\mathbf{N} = \mu^2 - 1 = -\tfrac{1}{2}\) | B1 | 2.1 |
| so \(\det\mathbf{N} = 1/\det\mathbf{M}\) | B1 | 2.2a |
| [2] |
Notes
(b)(i)
B1: oe eg \(\det\mathbf{N} = 1/\det\mathbf{M}\)
‘inverse’ B0
(b)(ii)
B1: soi
M1: elimination
or \(\lambda = -1/\mu,\ -1/\mu + 2\mu = 0\)
A1: or \(\mu = -\sqrt{2}/2\)
must be exact