Matrix algebra and linear transformationsEdexcel A-Level Further Maths: Subtopic test

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

   (a)

   Find $\mathbf{AB}$.

   [1 mark]

   • A$\begin{pmatrix}2&-2\\4&2\end{pmatrix}$
   • B$\begin{pmatrix}2&-1\\0&0\end{pmatrix}$
   • C$\begin{pmatrix}4&-2\\6&0\end{pmatrix}$
   • D$\begin{pmatrix}4&6\\-2&0\end{pmatrix}$

   (b)

   Find $\mathbf{BA}$.

   [1 mark]

   • A$\begin{pmatrix}2&-2\\4&2\end{pmatrix}$
   • B$\begin{pmatrix}4&-2\\6&0\end{pmatrix}$
   • C$\begin{pmatrix}2&-1\\0&0\end{pmatrix}$
   • D$\begin{pmatrix}2&4\\-2&2\end{pmatrix}$

   (c)

   Find the matrix $\mathbf{C}$ such that $\mathbf{A}+\mathbf{C}=2\mathbf{B}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Transformation $\mathrm{P}$ is a reflection in the line $y=x$. Transformation $\mathrm{Q}$ is a rotation through $90^\circ$ anticlockwise about the origin. The matrix $\mathbf{M}$ represents $\mathrm{P}$ followed by $\mathrm{Q}$.

   (a)

   Which matrix represents $\mathrm{Q}$?

   [1 mark]

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

   (b)

   Find the image of the point $(3,2)$ under $\mathbf{M}$.

   [1 mark]

   • A$(2,3)$
   • B$(-3,2)$
   • C$(3,-2)$
   • D$(-2,3)$

   (c)

   Find $\mathbf{M}$ and describe fully the single transformation represented by $\mathbf{M}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The matrix $\mathbf{T}$ represents an enlargement with scale factor $3$ and centre the origin, followed by a stretch with scale factor $2$ parallel to the $x$-axis.

   (a)

   Find $\mathbf{T}$.

   [3 marks]

   (b)

   The matrix $\mathbf{U}=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ represents a rotation through $90^\circ$ anticlockwise about the origin. Find the matrix that represents $\mathbf{T}$ followed by $\mathbf{U}$, and hence find the image of the point $(2,-1)$ under this combined transformation.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The matrix $\mathbf{M}=\begin{pmatrix}\frac12&-\frac{\sqrt3}{2}\\ \frac{\sqrt3}{2}&\frac12\end{pmatrix}$ represents a rotation about the origin.

   (a)

   (i) Find the angle of rotation represented by $\mathbf{M}$.
   (ii) Find $\mathbf{M}^2$.
   (iii) Find the smallest positive integer $n$ for which $\mathbf{M}^n=\mathbf{I}$, giving a reason.

   [6 marks]

   (b)

   The matrix $\mathbf{P}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ represents a reflection in the $x$-axis. Find $\mathbf{PM}$ and $\mathbf{MP}$, and explain what the result shows about combining a rotation with a reflection.

   [6 marks]

   Total for question 4: 12 marks

End of questions