FP1 June 2016 Q1
1. Given that \(k\) is a real number and that \[\mathbf{A} = \begin{pmatrix} 1 + k & k \\ k & 1 - k \end{pmatrix}\] find the exact values of \(k\) for which \(\mathbf{A}\) is a singular matrix. Give your answers in their simplest form. (3)
| Scheme | Marks |
|---|---|
| Determinant of \(\mathbf{A} = (1 - k)(1 + k) - k^2 = 0\) | M1 |
| \(1 - k + k - k^2 - k^2\ (= 0)\) \(1 - 2k^2\ (= 0)\) | A1 |
| So \(k = \dfrac{\pm\sqrt{2}}{2}\) | A1 |
| (3) | |
| (3 marks) |
Notes
M1: for attempting \(ad - bc = 0\) with ‘\(= 0\)’ seen or used later in the solution.
A1: Correct (unsimplified) expression on LHS or correct equation after brackets expanded.
A1: Accept \(\pm\dfrac{\sqrt{2}}{2}\), \(\pm\dfrac{1}{\sqrt{2}}\), \(\pm\sqrt{\dfrac{1}{2}}\), \(\pm\sqrt{0.5}\). Must have \(\pm\) for mark.