| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Find P and D for diagonalization / matrix powers |
| Difficulty | Challenging +1.2 This is a structured Further Maths diagonalization question with clear guidance at each step. While it involves multiple techniques (matrix inversion, eigenvalue verification, and expressing M^n using diagonalization), students are explicitly told what to verify and show. The computational work is substantial but routine for FP2 level, with no novel insights required—just systematic application of standard diagonalization methods. |
| Spec | 4.03i Determinant: area scale factor and orientation4.03j Determinant 3x3: calculation4.03l Singular/non-singular matrices4.03m det(AB) = det(A)*det(B)4.03o Inverse 3x3 matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Find cofactors/adjugate of \(\mathbf{P}\) | M1 | |
| \(\det(\mathbf{P}) = 4(-1-0)-2(-1-3)+k(0-1)= -4+8-k=4-k\) | M1, A1 | Correct determinant |
| \(\mathbf{P}^{-1} = \frac{1}{4-k}\begin{pmatrix}-1&2&4\\4&-4-k&-12+4k\\-1&2&2\end{pmatrix}\) (form) | M1 | Adjugate/det |
| When \(k=2\): \(\det=2\), giving stated matrix | A1, A1 | Verification |
| Answer | Marks | Guidance |
|---|---|---|
| \(\mathbf{M}\begin{pmatrix}4\\1\\1\end{pmatrix} = \lambda_1\begin{pmatrix}4\\1\\1\end{pmatrix}\): compute product | M1 | |
| \(= \begin{pmatrix}4\\-1+3-2\\8-3-2\end{pmatrix}\)... check each vector | A1 | |
| Eigenvalues: \(\lambda=0,\ \lambda=2,\ \lambda=-1\) (or as computed) | A1, A1 | One mark each eigenvalue |
| Answer | Marks | Guidance |
|---|---|---|
| Write \(\mathbf{M} = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}\) using diagonal matrix of eigenvalues | M1 | Diagonalisation method |
| \(\mathbf{M}^n = \mathbf{P}\mathbf{D}^n\mathbf{P}^{-1}\) | M1 | |
| \(\mathbf{D}^n = \text{diag}(0^n, 2^n, (-1)^n)\) | M1 | |
| Multiply out correctly | M1, M1 | |
| Separate into two matrices as required | A1, A1, A1 |
## Question 3:
**Part (i):** Find $\mathbf{P}^{-1}$, verify when $k=2$
| Find cofactors/adjugate of $\mathbf{P}$ | M1 | |
| $\det(\mathbf{P}) = 4(-1-0)-2(-1-3)+k(0-1)= -4+8-k=4-k$ | M1, A1 | Correct determinant |
| $\mathbf{P}^{-1} = \frac{1}{4-k}\begin{pmatrix}-1&2&4\\4&-4-k&-12+4k\\-1&2&2\end{pmatrix}$ (form) | M1 | Adjugate/det |
| When $k=2$: $\det=2$, giving stated matrix | A1, A1 | Verification |
**Part (ii):** Verify eigenvectors of $\mathbf{M}$, find eigenvalues
| $\mathbf{M}\begin{pmatrix}4\\1\\1\end{pmatrix} = \lambda_1\begin{pmatrix}4\\1\\1\end{pmatrix}$: compute product | M1 | |
| $= \begin{pmatrix}4\\-1+3-2\\8-3-2\end{pmatrix}$... check each vector | A1 | |
| Eigenvalues: $\lambda=0,\ \lambda=2,\ \lambda=-1$ (or as computed) | A1, A1 | One mark each eigenvalue |
**Part (iii):** Show $\mathbf{M}^n = \begin{pmatrix}4&-6&-10\\2&-3&-5\\0&0&0\end{pmatrix} + 2^{n-1}\begin{pmatrix}-2&4&4\\-3&6&6\\1&-2&-2\end{pmatrix}$
| Write $\mathbf{M} = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}$ using diagonal matrix of eigenvalues | M1 | Diagonalisation method |
| $\mathbf{M}^n = \mathbf{P}\mathbf{D}^n\mathbf{P}^{-1}$ | M1 | |
| $\mathbf{D}^n = \text{diag}(0^n, 2^n, (-1)^n)$ | M1 | |
| Multiply out correctly | M1, M1 | |
| Separate into two matrices as required | A1, A1, A1 | |
---3 Let $\mathbf { P } = \left( \begin{array} { r r r } 4 & 2 & k \\ 1 & 1 & 3 \\ 1 & 0 & - 1 \end{array} \right) ($ where $k \neq 4 )$ and $\mathbf { M } = \left( \begin{array} { r r r } 2 & - 2 & - 6 \\ - 1 & 3 & 1 \\ 1 & - 2 & - 2 \end{array} \right)$.\\
(i) Find $\mathbf { P } ^ { - 1 }$ in terms of $k$, and show that, when $k = 2 , \mathbf { P } ^ { - 1 } = \frac { 1 } { 2 } \left( \begin{array} { r r r } - 1 & 2 & 4 \\ 4 & - 6 & - 10 \\ - 1 & 2 & 2 \end{array} \right)$.\\
(ii) Verify that $\left( \begin{array} { l } 4 \\ 1 \\ 1 \end{array} \right) , \left( \begin{array} { l } 2 \\ 1 \\ 0 \end{array} \right)$ and $\left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right)$ are eigenvectors of $\mathbf { M }$, and find the corresponding eigenvalues.\\
(iii) Show that $\mathbf { M } ^ { n } = \left( \begin{array} { r r r } 4 & - 6 & - 10 \\ 2 & - 3 & - 5 \\ 0 & 0 & 0 \end{array} \right) + 2 ^ { n - 1 } \left( \begin{array} { r r r } - 2 & 4 & 4 \\ - 3 & 6 & 6 \\ 1 & - 2 & - 2 \end{array} \right)$.
Section B (18 marks)
\hfill \mbox{\textit{OCR MEI FP2 2007 Q3 [18]}}