FP3 June 2009 Q3
3. \[\mathbf{M} = \begin{pmatrix} 6 & 1 & -1 \\ 0 & 7 & 0 \\ 3 & -1 & 2 \end{pmatrix}\]
(a) Show that 7 is an eigenvalue of the matrix \(\mathbf{M}\) and find the other two eigenvalues of \(\mathbf{M}\). (5)
(b) Find an eigenvector corresponding to the eigenvalue 7. (4)
| Scheme | Marks |
|---|---|
| \(\begin{vmatrix} 6-\lambda & 1 & -1 \\ 0 & 7-\lambda & 0 \\ 3 & -1 & 2-\lambda \end{vmatrix} = 0 \quad \therefore (6-\lambda)(7-\lambda)(2-\lambda) + 3(7-\lambda) = 0\) | M1 |
| \((7-\lambda) = 0\) verifies \(\lambda = 7\) is an eigenvalue (can be seen anywhere) | M1 |
| \(\therefore (7-\lambda)\left\{12 - 8\lambda + \lambda^2 + 3\right\} = 0 \quad \therefore (7-\lambda)\left\{\lambda^2 - 8\lambda + 15\right\} = 0\) | A1 |
| \(\therefore (7-\lambda)(\lambda - 5)(\lambda - 3) = 0\) and 3 and 5 are the other two eigenvalues | M1 A1 |
| (5) |
| Scheme | Marks |
|---|---|
| Set \(\begin{pmatrix} 6 & 1 & -1 \\ 0 & 7 & 0 \\ 3 & -1 & 2 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix} = 7\begin{pmatrix} x \\ y \\ z \end{pmatrix}\) or \(\begin{pmatrix} -1 & 1 & -1 \\ 0 & 0 & 0 \\ 3 & -1 & -5 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}\) | M1 |
| Solve \(-x + y - z = 0\) and \(3x - y - 5z = 0\) to obtain \(x = 3z\) or \(y = 4z\) and a second equation which can contain 3 variables | M1 A1 |
| Obtain eigenvector as \(3\mathbf{i} + 4\mathbf{j} + \mathbf{k}\) (or multiple) | A1 |
| (4) | |
| (9 marks) |