FP1 January 2009 Q7
7. Given that \(\mathbf{X} = \begin{pmatrix} 2 & a \\ -1 & -1 \end{pmatrix}\), where \(a\) is a constant, and \(a \neq 2\),
Given that \(\mathbf{X} + \mathbf{X}^{-1} = \mathbf{I}\), where \(\mathbf{I}\) is the \(2 \times 2\) identity matrix,
| Scheme | Marks |
|---|---|
| The determinant is \(a - 2\) | M1 |
| \(\mathbf{X}^{-1} = \dfrac{1}{a - 2}\begin{pmatrix} -1 & -a \\ 1 & 2 \end{pmatrix}\) | M1 A1 |
| (3) |
Notes
(a) Attempt \(ad - bc\) for first M1
\(\dfrac{1}{\det}\begin{pmatrix} -1 & -a \\ 1 & 2 \end{pmatrix}\) for second M1
| Scheme | Marks |
|---|---|
| \(\mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) | B1 |
| Attempt to solve \(2 - \dfrac{1}{a - 2} = 1\), or \(a - \dfrac{a}{a - 2} = 0\), or \(-1 + \dfrac{1}{a - 2} = 0\), or \(-1 + \dfrac{2}{a - 2} = 1\) | M1 |
| To obtain \(a = 3\) only | A1 cso |
| (3) | |
| [6] |
Alternatives for (b)
If they use \(\mathbf{X}^2 + \mathbf{I} = \mathbf{X}\) they need to identify \(\mathbf{I}\) for B1, then attempt to solve suitable equation for M1 and obtain \(a = 3\) for A1
If they use \(\mathbf{X}^2 + \mathbf{X}^{-1} = \mathbf{O}\), they can score the B1 then marks for solving
If they use \(\mathbf{X}^3 + \mathbf{I} = \mathbf{O}\) they need to identify \(\mathbf{I}\) for B1, then attempt to solve suitable equation for M1 and obtain \(a = 3\) for A1
Notes
(b) Final A1 for correct solution only