FP3: Further matrix algebraEdexcel International A Level Further Maths: Topic test

1. 1
   The matrix $\mathbf{M}=\begin{pmatrix} 5 & 2 \\ 3 & 4 \end{pmatrix}$.

   (a)

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

   [1 mark]

   • A$\lambda^2+9\lambda+14=0$
   • B$\lambda^2-9\lambda+26=0$
   • C$\lambda^2-9\lambda+14=0$
   • D$\lambda^2-14\lambda+9=0$

   (b)

   Which of the following is an eigenvector of $\mathbf{M}$ corresponding to the eigenvalue $7$?

   [1 mark]

   • A$\begin{pmatrix} 1 \\ 1 \end{pmatrix}$
   • B$\begin{pmatrix} 2 \\ -3 \end{pmatrix}$
   • C$\begin{pmatrix} 2 \\ 3 \end{pmatrix}$
   • D$\begin{pmatrix} 1 \\ -1 \end{pmatrix}$

   (c)

   Find an eigenvector of $\mathbf{M}$ corresponding to the eigenvalue $2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The matrix $\mathbf{A}=\begin{pmatrix} 2 & 1 & 0 \\ 0 & 1 & k \\ 1 & 0 & 3 \end{pmatrix}$, where $k$ is a constant.

   (a)

   Find $\det\mathbf{A}$.

   [1 mark]

   • A$6-k$
   • B$6+k$
   • C$6$
   • D$k-6$

   (b)

   The matrix $\mathbf{A}$ is singular. Find the value of $k$.

   [1 mark]

   • A$6$
   • B$0$
   • C$-\frac{1}{6}$
   • D$-6$

   (c)

   Given that $k=1$, find the image of the point $(1,2,-1)$ under the transformation represented by $\mathbf{A}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The symmetric matrix $\mathbf{S}=\begin{pmatrix} 5 & -2 \\ -2 & 2 \end{pmatrix}$.

   (a)

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

   [3 marks]

   (b)

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

   [4 marks]

   Total for question 3: 7 marks

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

   (a)

   Show that $-1$ is an eigenvalue of $\mathbf{B}$ and find the other two eigenvalues.

   [6 marks]

   (b)

   Find an orthogonal matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that $\mathbf{P}^{\mathrm{T}}\mathbf{B}\mathbf{P}=\mathbf{D}$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The matrix $\mathbf{R}=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ represents a reflection in the plane $x=y$. The matrix $\mathbf{E}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}$ represents a stretch.

   (a)

   Find the image of the point $(3,-1,4)$ under the reflection represented by $\mathbf{R}$.

   [1 mark]

   • A$(-3,1,4)$
   • B$(3,-1,-4)$
   • C$(1,-3,4)$
   • D$(-1,3,4)$

   (b)

   Find $\det\mathbf{R}$.

   [1 mark]

   • A$1$
   • B$-1$
   • C$0$
   • D$2$

   (c)

   Find the matrix that represents the reflection $\mathbf{R}$ followed by the stretch $\mathbf{E}$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The matrix $\mathbf{N}=\begin{pmatrix} 4 & 0 & 1 \\ 2 & 6 & 2 \\ 1 & 0 & 4 \end{pmatrix}$.

   (a)

   The vector $\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$ is an eigenvector of $\mathbf{N}$. Find the corresponding eigenvalue.

   [1 mark]

   • A$6$
   • B$5$
   • C$4$
   • D$3$

   (b)

   Find the eigenvalue of $\mathbf{N}$ corresponding to the eigenvector $\begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}$.

   [1 mark]

   • A$5$
   • B$-3$
   • C$3$
   • D$6$

   (c)

   Show that $\begin{pmatrix} 1 \\ -4 \\ 1 \end{pmatrix}$ is an eigenvector of $\mathbf{N}$ and state the corresponding eigenvalue.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The matrix $\mathbf{A}=\begin{pmatrix} 1 & 1 & 0 \\ 0 & 2 & 1 \\ 1 & 0 & 2 \end{pmatrix}$ represents a transformation $T$.

   (a)

   Find $\mathbf{A}^{-1}$.

   [3 marks]

   (b)

   The transformation $T$ maps a point $P$ to the point $(3,3,-1)$. Find the coordinates of $P$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The matrix $\mathbf{N}=\begin{pmatrix} 2 & a \\ b & 3 \end{pmatrix}$, where $a$ and $b$ are constants, has eigenvectors $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$.

   (a)

   Find the values of $a$ and $b$, and the two eigenvalues of $\mathbf{N}$.

   [6 marks]

   (b)

   Given that $a=-1$ and $b=-2$, use the eigenvectors to find the image of the point $(5,-1)$ under the transformation represented by $\mathbf{N}^2$.

   [6 marks]

   Total for question 8: 12 marks

End of questions