| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | January |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find eigenvalues of 3×3 matrix |
| Difficulty | Standard +0.3 This is a standard FP2 eigenvalue/eigenvector question covering routine techniques: computing a characteristic equation (determinant expansion), verifying an eigenvalue, finding a unit eigenvector, and applying Cayley-Hamilton theorem. While Further Maths content is inherently more advanced, these are textbook procedures with clear algorithms and no novel insight required, placing it slightly above average difficulty overall. |
| Spec | 4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation |
| Answer | Marks |
|---|---|
| M1 | Obtaining \(\det(M - \lambda I)\) |
| A1 | Any correct form |
| M1 | Simplification |
| A1 (ag) | www, but condone omission of \(= 0\) |
| Total | 4 |
| Answer | Marks |
|---|---|
| M1 | Factorising and obtaining a quadratic. If M0, give B1 for substituting \(\lambda = 3\) |
| A1 | Correct quadratic |
| M1 | Considering discriminant o.e. |
| A1 | Conclusion from correct evidence www |
| Total | 4 |
| Answer | Marks |
|---|---|
| M1 | Two independent equations |
| M1 | Obtaining a non-zero eigenvector |
| A1 | |
| B1 | |
| B1 | Must be a magnitude |
| Total | 5 |
| Answer | Marks |
|---|---|
| M1 | Use of Cayley-Hamilton Theorem |
| M1 | Multiplying by \(M^{-1}\) and rearranging |
| A1 | Must contain I |
| Total | 3 |
## (i)
$\det(M - \lambda I) = (1-\lambda)[(3-\lambda)(1-\lambda) + 8]$
$+ 4[2(1-\lambda) - 2] + 5[8 + (3-\lambda)]$
$= (1-\lambda)(\lambda^2 - 4\lambda + 11) - 8k + 55 - 5\lambda = 0$
$\Rightarrow \lambda^3 - 5\lambda^2 + 28\lambda - 66 = 0$
| | M1 | Obtaining $\det(M - \lambda I)$ |
| | A1 | Any correct form |
| | M1 | Simplification |
| | A1 (ag) | www, but condone omission of $= 0$ |
| **Total** | **4** | |
## (ii)
$\lambda^3 - 5\lambda^2 + 28\lambda - 66 = 0$
$\Rightarrow (\lambda - 3)(\lambda^2 - 2\lambda + 22) = 0$
$\lambda^2 - 2\lambda + 22 = 0 \Rightarrow b^2 - 4ac = -84$
so no other real eigenvalues
| | M1 | Factorising and obtaining a quadratic. If M0, give B1 for substituting $\lambda = 3$ |
| | A1 | Correct quadratic |
| | M1 | Considering discriminant o.e. |
| | A1 | Conclusion from correct evidence www |
| **Total** | **4** | |
## (iii)
$\lambda = 3 \Rightarrow \begin{pmatrix} -2 & -4 & 5 \\ 2 & 0 & -2 \\ -1 & 4 & -2 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$
$\Rightarrow -2x - 4y + 5z = 0$
$2x - 2z = 0$
$-x + 4y - 2z = 0$
$\Rightarrow x = z = k, y = \frac{3}{4}k$
$\Rightarrow$ eigenvector is $\begin{pmatrix} 4 \\ 3 \\ 4 \end{pmatrix}$
$\Rightarrow$ eigenvector with unit length is $v = \frac{1}{\sqrt{41}}\begin{pmatrix} 4 \\ 3 \\ 4 \end{pmatrix}$
Magnitude of $M^T v$ is $3^n$
| | M1 | Two independent equations |
| | M1 | Obtaining a non-zero eigenvector |
| | A1 | |
| | B1 | |
| | B1 | Must be a magnitude |
| **Total** | **5** | |
## (iv)
$\lambda^3 - 5\lambda^2 + 28\lambda - 66 = 0$
$\Rightarrow M^3 - 5M^2 + 28M - 66I = 0$
$\Rightarrow M^2 - 5M + 28I - 66M^{-1} = 0$
$\Rightarrow M^{-1} = \frac{1}{66}(M^2 - 5M + 28I)$
| | M1 | Use of Cayley-Hamilton Theorem |
| | M1 | Multiplying by $M^{-1}$ and rearranging |
| | A1 | Must contain I |
| **Total** | **3** | |
**TOTAL FOR QUESTION 3: 16**
---\begin{enumerate}[label=(\roman*)]
\item Show that the characteristic equation of the matrix
$$\mathbf{M} = \begin{pmatrix} 1 & -4 & 5 \\ 2 & 3 & -2 \\ -1 & 4 & 1 \end{pmatrix}$$
is $\lambda^3 - 5\lambda^2 + 28\lambda - 66 = 0$. [4]
\item Show that $\lambda = 3$ is an eigenvalue of $\mathbf{M}$, and determine whether or not $\mathbf{M}$ has any other real eigenvalues. [4]
\item Find an eigenvector, $\mathbf{v}$, of unit length corresponding to $\lambda = 3$.
State the magnitude of the vector $\mathbf{M}^n\mathbf{v}$, where $n$ is an integer. [5]
\item Using the Cayley-Hamilton theorem, obtain an equation for $\mathbf{M}^{-1}$ in terms of $\mathbf{M}^2$, $\mathbf{M}$ and $\mathbf{I}$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP2 2011 Q3 [16]}}