Matrices (A2)AQA A-Level Further Maths: Topic test

1. 1
   The matrix $\mathbf{P}=\begin{pmatrix}1&2&0\\0&1&3\\2&0&1\end{pmatrix}$ represents a transformation of three-dimensional space. Lengths are in centimetres.

   (a)

   Find $\det\mathbf{P}$.

   [1 mark]

   • A$-11$
   • B$13$
   • C$1$
   • D$-13$

   (b)

   A solid of volume $4\ \text{cm}^3$ is transformed by $\mathbf{P}$. Find the volume of its image.

   [1 mark]

   • A$13\ \text{cm}^3$
   • B$17\ \text{cm}^3$
   • C$52\ \text{cm}^3$
   • D$\frac{4}{13}\ \text{cm}^3$

   (c)

   A second matrix $\mathbf{Q}$ has $\det\mathbf{Q}=-2$. Find the volume scale factor of the combined transformation $\mathbf{PQ}$ and state its effect on orientation.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Three planes have equations $\Pi_1:2x-y+z=3$, $\Pi_2:x+y-z=0$ and $\Pi_3:x-2y+2z=k$, where $k$ is a constant.

   (a)

   For which value of $k$ do the three planes meet in a common line?

   [1 mark]

   • A$0$
   • B$-3$
   • C$6$
   • D$3$

   (b)

   When $k=5$, how do the three planes meet?

   [1 mark]

   • AThey meet at a single point
   • BThey form a triangular prism: no point lies on all three planes and no two planes are parallel
   • CThey meet in a common line
   • DThey are three parallel planes

   (c)

   Show that the determinant of the coefficient matrix of the three equations is zero.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The matrix $\mathbf{M}=\begin{pmatrix}5&4\\1&2\end{pmatrix}$ represents a linear transformation of the plane.

   (a)

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

   [3 marks]

   (b)

   Find an eigenvector corresponding to each eigenvalue, and hence write down the equations of the two invariant lines through the origin.

   [4 marks]

   Total for question 3: 7 marks

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

   (a)

   Find the eigenvalues and corresponding eigenvectors of $\mathbf{N}$, and write down matrices $\mathbf{U}$ and $\mathbf{D}$, with $\mathbf{D}$ diagonal, such that $\mathbf{N}=\mathbf{UDU}^{-1}$.

   [6 marks]

   (b)

   Find $\mathbf{U}^{-1}$ and hence find $\mathbf{N}^n$ for positive integers $n$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The matrix $\mathbf{T}=\begin{pmatrix}1&1&1\\1&1+x&1\\1&1&1+y\end{pmatrix}$, where $x$ and $y$ are real constants.

   (a)

   Use row operations to find $\det\mathbf{T}$.

   [1 mark]

   • A$xy$
   • B$x+y$
   • C$x+y+xy$
   • D$1+xy$

   (b)

   When $x=2$ and $y=-3$, $\mathbf{T}$ represents a transformation of three-dimensional space. Which statement is correct?

   [1 mark]

   • AVolumes are multiplied by $6$ and orientation is preserved
   • BVolumes are multiplied by $-6$ and orientation is reversed
   • CVolumes are multiplied by $6$ and orientation is reversed
   • DVolumes are multiplied by $\frac16$ and orientation is reversed

   (c)

   Given that $y=2x$, find the values of $x$ for which $\det\mathbf{T}=18$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The matrix $\mathbf{C}=\begin{pmatrix}2&1&4\\0&3&5\\0&0&-1\end{pmatrix}$.

   (a)

   Which list gives the eigenvalues of $\mathbf{C}$?

   [1 mark]

   • A$2$, $3$ and $-1$
   • B$2$, $3$ and $1$
   • C$-2$, $-3$ and $1$
   • D$1$, $4$ and $5$

   (b)

   Which statement explains why $\mathbf{C}$ can be diagonalised?

   [1 mark]

   • AIts determinant is non-zero
   • BIt is an upper triangular matrix
   • CIts eigenvalues are all positive
   • DIt has three distinct real eigenvalues, so it has three linearly independent eigenvectors

   (c)

   Find an eigenvector of $\mathbf{C}$ corresponding to the eigenvalue $3$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Three planes have equations $x+2y+z=6$, $x+3y+2z=11$ and $y+2z=8$. The matrix of coefficients is $\mathbf{A}=\begin{pmatrix}1&2&1\\1&3&2\\0&1&2\end{pmatrix}$.

   (a)

   Show that $\det\mathbf{A}=1$, and explain what this tells you about how the three planes meet.

   [3 marks]

   (b)

   Given that $\mathbf{A}^{-1}=\begin{pmatrix}4&-3&1\\-2&2&-1\\1&-1&1\end{pmatrix}$, solve the three equations.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The matrix $\mathbf{K}=\begin{pmatrix}5&2&0\\2&2&0\\0&0&3\end{pmatrix}$.

   (a)

   Find the eigenvalues of $\mathbf{K}$ and an eigenvector corresponding to each eigenvalue.

   [6 marks]

   (b)

   Write down matrices $\mathbf{U}$ and $\mathbf{D}$, with $\mathbf{D}$ diagonal, such that $\mathbf{K}=\mathbf{UDU}^{-1}$. Hence show that $\det(\mathbf{K}^n)=18^n$ for positive integers $n$, and explain why $\mathbf{K}^n$ is non-singular for every $n$.

   [6 marks]

   Total for question 8: 12 marks

End of questions