Inverse transformations | Year 11

01 / LEARN

Understand the idea.

Reverse the chain

To reverse successive invertible transformations, undo the last action first. If $A$ is followed by $B$ , the original matrix is $B A$ and the inverse is $A^{- 1} B^{- 1}$ . Each inverse then meets the factor it cancels.

Check the order algebraically: $A^{- 1} B^{- 1} B A = I$ . A matrix inverse is not obtained by taking entrywise reciprocals. Singular maps, such as projections, cannot be reversed for every input because they discard information.

Subtract a translation before undoing the matrix

For an affine rule $q = A p + t$ , first subtract the final translation and then undo $A$ : $p = A^{- 1} (q - t)$ . Applying $A^{- 1}$ to $q$ before subtracting $t$ generally changes the displacement and gives the wrong input.

A rotation about centre $c$ is written $q = c + R (p - c)$ . Its inverse uses the same centre and the opposite rotation angle: translate to that centre, rotate back, then translate to the original coordinate system.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Undo scale and rotation

First dilate by 2, then rotate $90^{\circ}$ counterclockwise. Find the inverse matrix.

Worked solution

1
Write the original transformation matrix.
$\begin{aligned} T & = (\begin{array}{cccccccccc} 0 & - 1 \\ 1 & 0 \end{array}) (\begin{array}{cccccccccc} 2 & 0 \\ 0 & 2 \end{array}) \\ = (\begin{array}{cccccccccc} 0 & - 2 \\ 2 & 0 \end{array}) \end{aligned}$
Dilation acts first and the counterclockwise rotation acts second.
2
Undo the rotation, then the dilation.
$T^{- 1} = \frac{1}{2} (\begin{array}{cccccccccc} 0 & 1 \\ - 1 & 0 \end{array})$
A clockwise quarter-turn reverses the rotation, and a factor of $\frac{1}{2}$ reverses the dilation.
3
Check the inverse product.
$\begin{aligned} T^{- 1} T & = \frac{1}{2} (\begin{array}{cccccccccc} 0 & 1 \\ - 1 & 0 \end{array}) (\begin{array}{cccccccccc} 0 & - 2 \\ 2 & 0 \end{array}) \\ = \frac{1}{2} (\begin{array}{cccccccccc} 2 & 0 \\ 0 & 2 \end{array}) \\ = I \end{aligned}$
The adjacent reverse actions restore every position vector.

Answer

$\frac{1}{2} (\begin{array}{cccccccccc} 0 & 1 \\ - 1 & 0 \end{array})$ .

EXAMPLE 02

Recover a point

The affine rule $q = (\begin{array}{cccccccccc} 2 & 1 \\ 0 & 1 \end{array}) p + (1, - 2)$ gives $q = (8, 1)$ . Find p.

Worked solution

1
Subtract the translation from the known image.
$A p = q - t = (\begin{array}{cccccccccc} 8 - 1 \\ 1 - (- 2) \end{array}) = (\begin{array}{cccccccccc} 7 \\ 3 \end{array})$
Translation was the last action, so it is undone first.
2
Calculate the inverse linear map.
$det A = 2, \begin{aligned} A^{- 1} & = \frac{1}{2} (\begin{array}{cccccccccc} 1 & - 1 \\ 0 & 2 \end{array}) \\ = (\begin{array}{cccccccccc} \frac{1}{2} & - \frac{1}{2} \\ 0 & 1 \end{array}) \end{aligned}$
The non-zero determinant permits unique recovery after subtracting the displacement.
3
Apply the inverse to the translated image.
$\begin{aligned} p & = A^{- 1} (\begin{array}{cccccccccc} 7 \\ 3 \end{array}) \\ = (\begin{array}{cccccccccc} \frac{7}{2} - \frac{3}{2} \\ 3 \end{array}) \\ = (\begin{array}{cccccccccc} 2 \\ 3 \end{array}) \end{aligned}$
The inverse acts on $q - t$ , rather than on the untranslated image.
4
Verify the recovered point.
$\begin{aligned} A p + t & = (\begin{array}{cccccccccc} 2 & 1 \\ 0 & 1 \end{array}) (\begin{array}{cccccccccc} 2 \\ 3 \end{array}) + (\begin{array}{cccccccccc} 1 \\ - 2 \end{array}) \\ = (\begin{array}{cccccccccc} 4 + 3 + 1 \\ 3 - 2 \end{array}) \\ = (\begin{array}{cccccccccc} 8 \\ 1 \end{array}) \end{aligned}$
The original affine rule returns the stated image.

Answer

$p = (2, 3)$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

What undoes reflection in the x-axis?

Show answer & reasoning

The same reflection.

The same reflection undoes reflection in the $x$ -axis. Its rule $(x, y) \mapsto (x, - y)$ applied twice gives $(x, - (- y)) = (x, y)$ , so its square is the identity.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Find the inverse of a dilation of factor 4 about the origin.
2 marks
Find the inverse of $(\begin{array}{cccccccccc} 0 & - 1 \\ 1 & 0 \end{array})$ .
2 marks
Express ${(A B C)}^{- 1}$ using inverse factors.
2 marks
Find $p$ if $2 p + (3, 1) = (9, - 3)$ .
2 marks
Why is reflection invertible but projection onto the same axis not?
2 marks
Rotate $(3, 1)$ through $90^{\circ}$ counterclockwise about $(1, 1)$ . Find the image and inverse move.
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find the inverse of a dilation of factor 4 about the origin.
2 marks
Undo the scale using its reciprocal.
$\frac{1}{4} I$ .
Find the inverse of $(\begin{array}{cccccccccc} 0 & - 1 \\ 1 & 0 \end{array})$ .
2 marks
Undo a counterclockwise quarter-turn with a clockwise one.
$(\begin{array}{cccccccccc} 0 & 1 \\ - 1 & 0 \end{array})$ .
Express ${(A B C)}^{- 1}$ using inverse factors.
2 marks
Reverse the product order.
Adjacent inverse pairs then cancel.
$C^{- 1} B^{- 1} A^{- 1}$ .
Find $p$ if $2 p + (3, 1) = (9, - 3)$ .
2 marks
Subtract $(3, 1)$ to obtain $(6, - 4)$ .
Divide by 2.
$p = (3, - 2)$ .
Why is reflection invertible but projection onto the same axis not?
2 marks
Reflection is one-to-one with non-zero determinant.
Projection is singular.
Reflection preserves information; projection deletes one component.
Rotate $(3, 1)$ through $90^{\circ}$ counterclockwise about $(1, 1)$ . Find the image and inverse move.
2 marks
Relative to the centre, the vector is $(2, 0)$ .
Rotate it to $(0, 2)$ , then add the centre.
Image $(1, 3)$ ; rotate clockwise about $(1, 1)$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: matrices and transformations; original lessons follow the approved school-sized chapter structure.

Official curriculum source ↗

In this lesson

Understand the idea.

Reverse the chain

Subtract a translation before undoing the matrix

See the method in action.

Undo scale and rotation

Write the original transformation matrix.

Undo the rotation, then the dilation.

Check the inverse product.

Recover a point

Subtract the translation from the known image.

Calculate the inverse linear map.

Apply the inverse to the translated image.

Verify the recovered point.

Try it yourself.

Check your understanding

Keep practising