Inverse transformations | Year 11

01 / LEARN

The idea, step by step

Reverse the chain

For invertible factors, ${(B A)}^{- 1} = A^{- 1} B^{- 1}$ . The last action B must be undone first. Multiplying $A^{- 1} B^{- 1} B A$ verifies this by cancelling adjacent inverse pairs.

An inverse does not mean taking the reciprocal of each entry. Singular maps, including coordinate projections, cannot be reversed for every input.

Subtract a translation before undoing the matrix

If $q = A p + t$ , solve for the original p by $p = A^{- 1} (q - t)$ . Subtracting the translation and applying the inverse in the other order generally gives the wrong answer.

A rotation about c has rule q=c+R(p-c). Its inverse rotates through the opposite angle about the same centre, with the same translate-rotate-translate pattern.

A common trap

Undoing actions in the original order.

Instead: Start with the last action and work backwards.

02 / WORKED EXAMPLES

See the thinking

Example 1

Undo scale and rotation

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

The original combined matrix is twice the counterclockwise quarter-turn matrix.
Undo the quarter-turn clockwise, then multiply both coordinates by one half.

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

Example 2

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.

Subtract the translation to obtain (7,3).
The inverse matrix is $(\begin{array}{cccccccccc} \frac{1}{2} & - \frac{1}{2} \\ 0 & 1 \end{array})$ .
Its product with (7,3) is (2,3).

p = (2, 3)

Check your understanding

What undoes reflection in the x-axis?

Show answer

The same reflection.

Reflecting twice restores every point.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF 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 ↗

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