Further Pure 2: Further matrix algebraEdexcel A-Level Further Maths: Topic test

1. 1
   The matrix $\mathbf{M}=\begin{pmatrix}1&2\\ 4&3\end{pmatrix}$ has real eigenvalues.

   (a)

   What is the characteristic equation of $\mathbf{M}$?

   [1 mark]

   • A$\lambda^2-4\lambda-5=0$
   • B$\lambda^2+4\lambda-5=0$
   • C$\lambda^2-4\lambda+5=0$
   • D$\lambda^2-4\lambda+11=0$

   (b)

   Which vector is an eigenvector of $\mathbf{M}$ corresponding to the eigenvalue $5$?

   [1 mark]

   • A$\begin{pmatrix}1\\ -2\end{pmatrix}$
   • B$\begin{pmatrix}2\\ 1\end{pmatrix}$
   • C$\begin{pmatrix}1\\ 2\end{pmatrix}$
   • D$\begin{pmatrix}1\\ -1\end{pmatrix}$

   (c)

   Find an eigenvector of $\mathbf{M}$ corresponding to the eigenvalue $-1$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The matrix $\mathbf{N}=\begin{pmatrix}2&-1\\ 3&6\end{pmatrix}$.

   (a)

   What is the characteristic equation of $\mathbf{N}$?

   [1 mark]

   • A$\lambda^2+8\lambda+15=0$
   • B$\lambda^2-8\lambda+15=0$
   • C$\lambda^2-8\lambda+9=0$
   • D$\lambda^2-8\lambda-15=0$

   (b)

   By the Cayley-Hamilton theorem, which expression is equal to $\mathbf{N}^3$?

   [1 mark]

   • A$8\mathbf{N}-15\mathbf{I}$
   • B$64\mathbf{N}-120\mathbf{I}$
   • C$49\mathbf{N}-15\mathbf{I}$
   • D$49\mathbf{N}-120\mathbf{I}$

   (c)

   Use the Cayley-Hamilton theorem to find $\mathbf{N}^{-1}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The matrix $\mathbf{B}=\begin{pmatrix}3&-5\\ 1&-1\end{pmatrix}$ has complex eigenvalues.

   (a)

   Find the eigenvalues of $\mathbf{B}$.

   [3 marks]

   (b)

   Find an eigenvector of $\mathbf{B}$ corresponding to each eigenvalue.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The symmetric matrix $\mathbf{S}=\begin{pmatrix}2&0&0\\ 0&3&1\\ 0&1&3\end{pmatrix}$.

   (a)

   Show that $\mathbf{S}$ has a repeated eigenvalue, and find the eigenvalues and corresponding eigenvectors.

   [6 marks]

   (b)

   Find an orthogonal matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that $\mathbf{P}^{\mathrm{T}}\mathbf{S}\mathbf{P}=\mathbf{D}$, and hence find $\mathbf{S}^4$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The matrix $\mathbf{A}=\begin{pmatrix}5&-2\\ 4&-1\end{pmatrix}$ has eigenvector $\begin{pmatrix}1\\ 2\end{pmatrix}$ with eigenvalue $1$ and eigenvector $\begin{pmatrix}1\\ 1\end{pmatrix}$ with eigenvalue $3$. The matrix $\mathbf{P}$ has these two eigenvectors as its columns, in the order given, so that $\mathbf{P}^{-1}\mathbf{A}\mathbf{P}=\mathbf{D}$ for a diagonal matrix $\mathbf{D}$.

   (a)

   What is $\mathbf{D}$?

   [1 mark]

   • A$\operatorname{diag}(3,1)$
   • B$\operatorname{diag}(1,3)$
   • C$\operatorname{diag}(1,2)$
   • D$\operatorname{diag}(5,-1)$

   (b)

   What are the eigenvalues of $\mathbf{A}^4$?

   [1 mark]

   • A$1$ and $81$
   • B$4$ and $12$
   • C$1$ and $12$
   • D$1$ and $9$

   (c)

   Write down $\mathbf{P}^{-1}$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The matrix $\mathbf{K}=\begin{pmatrix}4&-1\\ 1&2\end{pmatrix}$.

   (a)

   What is the repeated eigenvalue of $\mathbf{K}$?

   [1 mark]

   • A$1$
   • B$2$
   • C$6$
   • D$3$

   (b)

   Which vector is an eigenvector of $\mathbf{K}$?

   [1 mark]

   • A$\begin{pmatrix}1\\ -1\end{pmatrix}$
   • B$\begin{pmatrix}1\\ 3\end{pmatrix}$
   • C$\begin{pmatrix}1\\ 1\end{pmatrix}$
   • D$\begin{pmatrix}4\\ -1\end{pmatrix}$

   (c)

   Explain why $\mathbf{K}$ cannot be diagonalised.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The matrix $\mathbf{T}=\begin{pmatrix}1&1&0\\ 0&2&1\\ 0&0&3\end{pmatrix}$.

   (a)

   Show that the characteristic equation of $\mathbf{T}$ is $\lambda^3-6\lambda^2+11\lambda-6=0$.

   [3 marks]

   (b)

   Use the Cayley-Hamilton theorem to find $\mathbf{T}^{-1}$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The matrix $\mathbf{M}=\begin{pmatrix}0&1\\ -2&3\end{pmatrix}$.

   (a)

   Find the eigenvalues of $\mathbf{M}$ and a normalised eigenvector for each.

   [6 marks]

   (b)

   Find $\mathbf{M}^n$ in terms of $n$, and verify your result for $n=2$ using the Cayley-Hamilton theorem.

   [6 marks]

   Total for question 8: 12 marks

End of questions