Linear transformations as matricesEdexcel International A Level Further Maths: Subtopic test

1. 1
   A linear transformation $T$ of the plane maps $(1,0)$ to $(2,3)$ and $(0,1)$ to $(-1,4)$.

   (a)

   Write down the matrix that represents $T$.

   [1 mark]

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

   (b)

   Find the image of the point $(3,2)$ under $T$.

   [1 mark]

   • A$(12,5)$
   • B$(4,17)$
   • C$(5,7)$
   • D$(17,4)$

   (c)

   Find the image of the point $(-2,5)$ under $T$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The matrix $\mathbf{R}=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}$ represents the transformation $R$ and the matrix $\mathbf{S}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$ represents the transformation $S$.

   (a)

   Find the image of the point $(3,2)$ under $R$.

   [1 mark]

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

   (b)

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

   [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$\begin{pmatrix}0&-1\\ -1&0\end{pmatrix}$

   (c)

   Find the image of the point $(4,1)$ under $S$ followed by $R$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The linear transformation $T$ is represented by $\mathbf{T}=\begin{pmatrix}2&1\\ -1&3\end{pmatrix}$ and the linear transformation $U$ is represented by $\mathbf{U}=\begin{pmatrix}1&2\\ 0&-1\end{pmatrix}$.

   (a)

   Find the matrix that represents $T$ followed by $U$, and use it to find the image of the point $(1,2)$ under this combined transformation.

   [3 marks]

   (b)

   Find the matrix that represents $U$ followed by $T$. By considering where each combined transformation sends the point $(1,0)$, show that $T$ followed by $U$ is not the same transformation as $U$ followed by $T$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A linear transformation $M$ of the plane has matrix $\mathbf{M}=\begin{pmatrix}a&b\\ c&d\end{pmatrix}$, where $a$, $b$, $c$ and $d$ are constants. Under $M$ the point $(1,2)$ maps to $(5,3)$ and the point $(2,-1)$ maps to $(5,-4)$.

   (a)

   Find the values of $a$, $b$, $c$ and $d$.

   [6 marks]

   (b)

   The linear transformation $N$ has matrix $\mathbf{N}=\begin{pmatrix}2&0\\ 1&1\end{pmatrix}$.
   (i) Find the matrix that represents $N$ followed by $M$.
   (ii) Hence find the image of the point $(2,-3)$ under $N$ followed by $M$.
   (iii) Find the matrix that represents $M$ followed by $N$, and state whether the two transformations are the same.

   [6 marks]

   Total for question 4: 12 marks

End of questions