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

1. 1
   The matrices $\mathbf{A}=\begin{pmatrix}3&1\\ -2&0\end{pmatrix}$ and $\mathbf{B}=\begin{pmatrix}1&4\\ 2&-1\end{pmatrix}$.

   (a)

   Find $\mathbf{AB}$.

   [1 mark]

   • A$\begin{pmatrix}-5&1\\ 8&2\end{pmatrix}$
   • B$\begin{pmatrix}3&4\\ -4&0\end{pmatrix}$
   • C$\begin{pmatrix}5&11\\ -2&-8\end{pmatrix}$
   • D$\begin{pmatrix}4&5\\ 0&-1\end{pmatrix}$

   (b)

   Find $2\mathbf{A}-\mathbf{B}$.

   [1 mark]

   • A$\begin{pmatrix}5&-2\\ -6&1\end{pmatrix}$
   • B$\begin{pmatrix}7&6\\ -2&-1\end{pmatrix}$
   • C$\begin{pmatrix}1&-7\\ -6&2\end{pmatrix}$
   • D$\begin{pmatrix}4&-6\\ -8&2\end{pmatrix}$

   (c)

   Find $\mathbf{BA}$, and state whether $\mathbf{AB}=\mathbf{BA}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   In the plane, $S$ is the reflection in the line $y=-x$ and $T$ is a stretch of scale factor $3$ parallel to the $x$-axis, with the $y$-axis invariant. The matrices of $S$ and $T$ are $\mathbf{S}$ and $\mathbf{T}$.

   (a)

   Which matrix is $\mathbf{S}$?

   [1 mark]

   • A$\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$
   • B$\begin{pmatrix}-1&0\\ 0&-1\end{pmatrix}$
   • C$\begin{pmatrix}-1&0\\ 0&1\end{pmatrix}$
   • D$\begin{pmatrix}0&-1\\ -1&0\end{pmatrix}$

   (b)

   Which matrix represents $S$ followed by $T$?

   [1 mark]

   • A$\begin{pmatrix}0&-1\\ -3&0\end{pmatrix}$
   • B$\begin{pmatrix}0&-3\\ -1&0\end{pmatrix}$
   • C$\begin{pmatrix}0&3\\ 1&0\end{pmatrix}$
   • D$\begin{pmatrix}3&-1\\ -1&1\end{pmatrix}$

   (c)

   Find the coordinates of the image of the point $(2,5)$ after $S$ followed by $T$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The matrix $\mathbf{M}=\begin{pmatrix}k&4\\ 1&k\end{pmatrix}$, where $k$ is a constant.

   (a)

   Find the values of $k$ for which $\mathbf{M}$ is singular.

   [3 marks]

   (b)

   When $k=3$, find $\mathbf{M}^{-1}$, and hence find the values of $x$ and $y$ such that $\mathbf{M}\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}10\\ 5\end{pmatrix}$.

   [4 marks]

   Total for question 3: 7 marks

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

   (a)

   Find the invariant points of $T$, and find the equations of the two invariant lines of $T$ that pass through the origin.

   [6 marks]

   (b)

   (i) The triangle $OAB$ has vertices $O(0,0)$, $A(1,0)$ and $B(0,2)$. Find the area of the image of $OAB$ under $T$. (ii) Find $\mathbf{T}^{-1}$, and hence find the point whose image under $T$ is $(5,-3)$.

   [6 marks]

   Total for question 4: 12 marks

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

   (a)

   Find $\det\mathbf{N}$.

   [1 mark]

   • A$13$
   • B$3$
   • C$-3$
   • D$8$

   (b)

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

   [1 mark]

   • A$\frac13\begin{pmatrix}2&5\\ 1&4\end{pmatrix}$
   • B$\frac13\begin{pmatrix}4&-5\\ -1&2\end{pmatrix}$
   • C$\begin{pmatrix}2&-5\\ -1&4\end{pmatrix}$
   • D$\frac13\begin{pmatrix}2&-5\\ -1&4\end{pmatrix}$

   (c)

   The matrix $\mathbf{P}$ is such that $\det(\mathbf{NP})=-12$. Find $\det\mathbf{P}$ and $\det\left(\mathbf{N}^{-1}\right)$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The matrix $\mathbf{S}=\begin{pmatrix}1&3\\ 0&1\end{pmatrix}$ represents a shear $S$ of the plane.

   (a)

   Which describes the set of invariant points of $S$?

   [1 mark]

   • AAll points on the $x$-axis
   • BOnly the origin
   • CAll points on the $y$-axis
   • DAll points on the line $y=x$

   (b)

   Which line is mapped onto itself by $S$ but is not a line of invariant points?

   [1 mark]

   • A$x=2$
   • B$y=x$
   • C$y=2$
   • D$y=-x$

   (c)

   Find the matrix of the inverse transformation $S^{-1}$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   In three dimensions, $R$ is the reflection in the plane $z=0$ and $Q$ is the rotation through $90^\circ$ anticlockwise about the $z$-axis, as seen looking from the positive $z$-axis towards the origin.

   (a)

   Write down the $3\times3$ matrices $\mathbf{R}$ and $\mathbf{Q}$ that represent $R$ and $Q$.

   [3 marks]

   (b)

   Find the matrix that represents $R$ followed by $Q$, and hence find the image of the point $(2,-1,3)$ under $R$ followed by $Q$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The matrices $\mathbf{U}=\begin{pmatrix}3&1\\ 5&2\end{pmatrix}$ and $\mathbf{V}=\begin{pmatrix}1&2\\ -1&0\end{pmatrix}$, and $\mathbf{I}$ is the $2\times2$ identity matrix.

   (a)

   Find $\mathbf{U}^{-1}$, and hence find the matrix $\mathbf{X}$ such that $\mathbf{UX}=\mathbf{V}$.

   [6 marks]

   (b)

   Find the values of $k$ for which $\mathbf{U}+k\mathbf{I}$ is singular, giving your answers in exact form.

   [6 marks]

   Total for question 8: 12 marks

End of questions