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Matrix Transformations

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen and paper handy, to work out your answers.

Question 11 mark

Which transformation is represented by the matrix \begin{pmatrix}0&1\\-1&0\end{pmatrix} ?

Select the correct answer.

Hint

The columns of the matrix are the images of (1, 0) and (0, 1).

Worked solution
  1. First column: (1, 0) maps to (0, -1)
  2. Second column: (0, 1) maps to (1, 0)
  3. Both points have turned a quarter turn clockwise about O.
  4. Rotation 90^\circ clockwise about the origin

Question 22 marks

Matrix \mathbf{R} represents a reflection in the line y=-x

\mathbf{R} maps the point P to the point (3, -7)

Work out the coordinates of P.

Write your answer as (x, y)

Hint

Find \mathbf{R} by working out where (1, 0) and (0, 1) go, then write \mathbf{R}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}3\\-7\end{pmatrix}.

Worked solution
  1. (1, 0)\to(0, -1) and (0, 1)\to(-1, 0), so \mathbf{R}=\begin{pmatrix}0&-1\\-1&0\end{pmatrix}
  2. Let P=(x, y): \;\begin{pmatrix}0&-1\\-1&0\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}-y\\-x\end{pmatrix}
  3. -y=3 so y=-3
  4. -x=-7 so x=7
  5. P=(7, -3)

Question 3Challenge5 marks

Matrix \mathbf{A} represents a rotation of 270^\circ anticlockwise about the origin.

Matrix \mathbf{B} represents a reflection in the x-axis.

A shape is transformed by \mathbf{A} followed by \mathbf{B}.

(a)

Use matrix multiplication to work out the image of the point (5, -2) under the combined transformation.

2 marks

Write your answer as (x, y)

(b)

Which single transformation is the same as \mathbf{A} followed by \mathbf{B}?

Select the correct answer.

1 mark

(c)

Point Q is transformed by \mathbf{A} followed by \mathbf{B}, and the image is then enlarged by scale factor 2, centre the origin.

The final image is (8, -6)

Work out the coordinates of Q.

2 marks

Write your answer as (x, y)

Hint

Write down both matrices by tracking (1, 0) and (0, 1). The transformation that happens second goes on the LEFT when you multiply.

Worked solution

Part (a)

  1. (1, 0)\to(0, -1) and (0, 1)\to(1, 0), so \mathbf{A}=\begin{pmatrix}0&1\\-1&0\end{pmatrix}
  2. \mathbf{B}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}
  3. \mathbf{A} first, so the combined matrix is \mathbf{BA} (second transformation on the left).
  4. \mathbf{BA}=\begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}0&1\\-1&0\end{pmatrix}=\begin{pmatrix}0&1\\1&0\end{pmatrix}
  5. \begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}5\\-2\end{pmatrix}=\begin{pmatrix}-2\\5\end{pmatrix}
  6. Image is (-2, 5)

Part (b)

  1. \mathbf{BA}=\begin{pmatrix}0&1\\1&0\end{pmatrix} swaps the coordinates: (1, 0)\to(0, 1) and (0, 1)\to(1, 0)
  2. Reflection in the line y=x
  3. (Reflection in y=-x comes from multiplying in the wrong order, \mathbf{AB}.)

Part (c)

  1. The enlargement matrix is \begin{pmatrix}2&0\\0&2\end{pmatrix}
  2. Overall matrix: \;\begin{pmatrix}2&0\\0&2\end{pmatrix}\begin{pmatrix}0&1\\1&0\end{pmatrix}=\begin{pmatrix}0&2\\2&0\end{pmatrix}
  3. Let Q=(x, y): \;\begin{pmatrix}0&2\\2&0\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}2y\\2x\end{pmatrix}=\begin{pmatrix}8\\-6\end{pmatrix}
  4. 2y=8 so y=4; 2x=-6 so x=-3
  5. Q=(-3, 4)

More on this topic: Matrix Transformations worksheet with full solutions

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