Combined transformations and their order | Year 11

01 / LEARN

The idea, step by step

First action, rightmost factor

If A acts first and B second, the final vector is $B (A p) = (B A) p$ . Write an arrow chain before multiplying. For three actions A, then B, then C, the combined matrix is CBA.

Associativity allows different brackets in a compatible product, not a change in the factor order.

When do maps commute?

A scalar dilation kI commutes with every linear transformation because $A (k I) = (k I) A = k A$ . In contrast, reflection and rotation often do not commute.

One vector with different outputs disproves a claim that two transformations commute. To show they commute for every input, compare the full matrices or establish an identity.

A common trap

Writing AB because the instruction says A then B.

Instead: For column vectors, the rightmost factor acts first.

02 / WORKED EXAMPLES

See the thinking

Example 1

Reflect then rotate

First reflect in the x-axis, then rotate $90^{\circ}$ counterclockwise. Find the matrix.

Call the two matrices F and R. The combined map is RF.
Multiplication gives $(\begin{array}{cccccccccc} 0 & 1 \\ 1 & 0 \end{array})$ , which swaps coordinates.

Reflection in

y = x

Example 2

Reverse the order

Apply these actions to $(1, 2)$ in both orders.

F then R sends (1,2) to (1,-2), then (2,1).
R then F sends (1,2) to (-2,1), then (-2,-1).

Outputs

(2, 1)

and

(- 2, - 1)

differ.

Check your understanding

A is followed by B and then C. Give the combined matrix.

Show answer

CBA.

Read the action on a column vector from right to left.

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

First dilate by factor 3, then reflect in the y-axis. Give the matrix.
2 marks
First reflect in $y = x$ , then in the x-axis. Describe the result.
2 marks
For $A = (\begin{array}{cccccccccc} 2 & 0 \\ 0 & 1 \end{array})$ and $B = (\begin{array}{cccccccccc} 0 & - 1 \\ 1 & 0 \end{array})$ , find BA.
2 marks
For those A and B, find AB and compare.
2 marks
Explain why dilation kI commutes with every linear transformation A.
2 marks
Project $(2, 5)$ onto the x-axis, then rotate $90^{\circ}$ counterclockwise.
2 marks

Read the worked solutions

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

First dilate by factor 3, then reflect in the y-axis. Give the matrix.
2 marks
Multiply the reflection matrix by 3I.
$(\begin{array}{cccccccccc} - 3 & 0 \\ 0 & 3 \end{array})$ .
First reflect in $y = x$ , then in the x-axis. Describe the result.
2 marks
The coordinate rule is (x,y) to (y,x), then (y,-x).
Rotation $90^{\circ}$ clockwise.
For $A = (\begin{array}{cccccccccc} 2 & 0 \\ 0 & 1 \end{array})$ and $B = (\begin{array}{cccccccccc} 0 & - 1 \\ 1 & 0 \end{array})$ , find BA.
2 marks
Use row-column multiplication in the written order.
$(\begin{array}{cccccccccc} 0 & - 1 \\ 2 & 0 \end{array})$ .
For those A and B, find AB and compare.
2 marks
The off-diagonal entries are different.
$A B = (\begin{array}{cccccccccc} 0 & - 2 \\ 1 & 0 \end{array}) \neq B A$ .
Explain why dilation kI commutes with every linear transformation A.
2 marks
The scalar multiplies each entry of A on either side.
$A (k I) = (k I) A = k A$ .
Project $(2, 5)$ onto the x-axis, then rotate $90^{\circ}$ counterclockwise.
2 marks
Projection gives (2,0).
The rotation gives (0,2).
$(0, 2)$ .

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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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