Teach Me Maths

Matrix Transformations

How to build and recognise transformation matrices, transform the unit square and combine transformations in the correct order.

The columns tell you where the unit steps go

A transformation matrix acts on a column vector:

M\binom{x}{y}=\binom{x'}{y'}

Its first column is the image of (1,0); its second is the image of (0,1)

The rotations and enlargements here are centred on the origin. Every two-by-two transformation matrix fixes the origin, so a non-zero translation cannot be represented in this form.

Worked example 1

Find the matrix for a rotation of 90° anticlockwise about the origin.

  1. (1,0) moves to (0,1), so the first column is \binom01. Also (0,1) moves to (-1,0)

  2. Your turn. What is the top entry of the second column?

    The top-right entry is -1

  3. Answer: R=\begin{pmatrix}0&-1\\1&0\end{pmatrix}. Put the image vectors in columns, not rows.

Explore the image of the unit square

The original vertices are A=(0,0), B=(1,0), C=(1,1), D=(0,1). Follow the labels as well as the outline: a reflection can leave the same square region while swapping vertices.

Recognise the standard matrices

Anticlockwise rotations about the origin

For 90^\circ:

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

For 180^\circ:

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

For 270^\circ:

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

Reflections in the axes

In the x-axis:

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

In the y-axis:

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

Reflections in diagonal lines

In y=x:

\begin{pmatrix}0&1\\1&0\end{pmatrix}

In y=-x:

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

Enlargement of scale factor k:

\begin{pmatrix}k&0\\0&k\end{pmatrix}

Worked example 2

Find the image of P=(-3,2) after reflection in the y-axis.

  1. Use \begin{pmatrix}-1&0\\0&1\end{pmatrix}\begin{pmatrix}-3\\2\end{pmatrix}. Reflection in the y-axis reverses the x coordinate and preserves the y coordinate.

  2. Your turn. What is the image’s x coordinate?

    The new x coordinate is 3

  3. Answer: P'=(3,2). State the mirror line when describing a reflection.

Worked example 3

An enlargement has centre (0,0) and scale factor -2. Find its matrix and the image of (2,-3).

  1. The matrix is \begin{pmatrix}-2&0\\0&-2\end{pmatrix}. Multiply both coordinates by -2

  2. Your turn. Find the image’s x coordinate.

    The image’s x coordinate is -4

  3. Answer: matrix \begin{pmatrix}-2&0\\0&-2\end{pmatrix}, image (-4,6). A negative scale factor puts each image point on the opposite side of the centre.

Worked example 4

Describe the transformation \begin{pmatrix}0&1\\1&0\end{pmatrix} and find the image of (4,-2).

  1. The matrix sends (x,y) to (y,x): it swaps the coordinates. This is reflection in y=x

  2. Your turn. What is the image’s y coordinate?

    The image’s y coordinate is 4

  3. Answer: reflection in y=x; the image is (-2,4)

Worked example 5

Describe the transformation with matrix \begin{pmatrix}0&1\\-1&0\end{pmatrix}

  1. The first column sends (1,0) to (0,-1). The second sends (0,1) to (1,0)

  2. Your turn. Under this transformation, what is the x coordinate of the image of (2,3)?

    The full image is (3,-2)

  3. Answer: rotation 90^\circ clockwise, equivalently 270^\circ anticlockwise, about the origin. Include the centre in a complete description.

The rightmost matrix acts first

If R is followed by S, the combined matrix is SR:

S(R\mathbf v)=(SR)\mathbf v

The order in the written product is the reverse of the order in which the point experiences the transformations.

Worked example 6

Let R=\begin{pmatrix}0&-1\\1&0\end{pmatrix} and S=\begin{pmatrix}1&0\\0&-1\end{pmatrix}. Find and describe the transformation R followed by S.

  1. The order is SR. Its top row is (0,-1)

  2. Your turn. Find the bottom-left entry of SR: 0(0)+(-1)(1)

    The bottom-left entry is -1

  3. Answer: SR=\begin{pmatrix}0&-1\\-1&0\end{pmatrix}, reflection in y=-x. For example, (2,1) goes to (-1,-2)

Worked example 7

Using the same R=\begin{pmatrix}0&-1\\1&0\end{pmatrix} and S=\begin{pmatrix}1&0\\0&-1\end{pmatrix}, find S followed by R and compare the image of (2,1).

  1. This order gives RS=\begin{pmatrix}0&1\\1&0\end{pmatrix}, reflection in y=x

  2. Your turn. After swapping the coordinates of (2,1), what is the new x coordinate?

    The image is (1,2)

  3. Answer: reflection in y=x, taking (2,1) to (1,2). Reversing the previous order changes both the matrix and the image.

Common mistakes

  • Putting images of unit steps in rows. Use their images as the first and second columns, in that order.
  • Reversing rotation direction. Trace a known point to check clockwise or anticlockwise.
  • Changing the wrong coordinate in a reflection. Reflection in the x-axis changes y; reflection in the y-axis changes x.
  • Giving an incomplete transformation description. Give the mirror line, or the centre plus rotation angle/direction or enlargement scale factor.
  • Multiplying transformations in spoken order. The rightmost matrix acts first: applying A then B gives BA.

Now try it: Matrix Transformations practice questions

More on this topic: Matrix Transformations worksheet with full solutions