Combined transformations and their order | Year 11

01 / LEARN

Understand the idea.

First action, rightmost factor

For column vectors, the rightmost matrix acts first. If $A$ acts and then $B$ , the output is $B (A p) = (B A) p$ . Write an arrow chain before multiplying to connect the verbal order with the factor order.

For three successive actions $A$ , then $B$ , then $C$ , the combined matrix is $C B A$ . Associativity permits changing brackets to reduce arithmetic, but it does not permit rearranging the factors.

When do maps commute?

Two maps commute when reversing their order gives the same result for every input. A scalar dilation $k I$ commutes with any linear map, since $A (k I) = (k I) A = k A$ . Other pairs require checking.

To disprove commutativity, one input giving two different outputs is sufficient. To prove it for all inputs, compare the full products $A B$ and $B A$ or establish a general identity; agreement at one point is insufficient.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Reflect then rotate

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

Worked solution

1
Write each standard matrix.
$F = (\begin{array}{cccccccccc} 1 & 0 \\ 0 & - 1 \end{array}), R = (\begin{array}{cccccccccc} 0 & - 1 \\ 1 & 0 \end{array})$
Reflection in the $x$ -axis reverses height; the rotation is a counterclockwise quarter-turn.
2
Put the first action on the right.
$p → F F p → R R F p$
The rotation must act on the already reflected vector.
3
Calculate the combined product.
$\begin{aligned} R F & = (\begin{array}{cccccccccc} 0 & - 1 \\ 1 & 0 \end{array}) (\begin{array}{cccccccccc} 1 & 0 \\ 0 & - 1 \end{array}) \\ = (\begin{array}{cccccccccc} 0 & 1 \\ 1 & 0 \end{array}) \end{aligned}$
Row-column multiplication gives a matrix that swaps the coordinates.
4
Identify its geometric action.
$R F (\begin{array}{cccccccccc} x \\ y \end{array}) = (\begin{array}{cccccccccc} y \\ x \end{array})$
Swapping coordinates is reflection in the line $y = x$ .

Answer

Reflection in $y = x$ .

EXAMPLE 02

Reverse the order

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

Worked solution

1
Reflect first, then rotate.
$(1, 2) → F (1, - 2) → R (2, 1)$
Reflection reverses $y$ , then rotation uses $(x, y) \mapsto (- y, x)$ .
2
Rotate first, then reflect.
$(1, 2) → R (- 2, 1) → F (- 2, - 1)$
The second order reverses the height after the quarter-turn.
3
Compare the final points.
$\begin{aligned} R F (\begin{array}{cccccccccc} 1 \\ 2 \end{array}) & = (\begin{array}{cccccccccc} 2 \\ 1 \end{array}) \\ F R (\begin{array}{cccccccccc} 1 \\ 2 \end{array}) & = (\begin{array}{cccccccccc} - 2 \\ - 1 \end{array}) \end{aligned}$
The final points differ, proving that these two transformations do not commute.

Answer

Outputs $(2, 1)$ and $(- 2, - 1)$ differ.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

CBA.

The vector follows $p \mapsto A p \mapsto B A p \mapsto C B A p$ . Therefore the combined matrix is $C B A$ ; the rightmost factor is the first action.

Keep practising

6 worksheet questions · 12 marks

Open worksheet 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 $B A$ .
2 marks
For those $A$ and $B$ , find $A B$ and compare.
2 marks
Explain why dilation $k I$ 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 $3 I$ .
$(\begin{array}{cccccccccc} - 3 & 0 \\ 0 & 3 \end{array})$ .
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 $B A$ .
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 $A B$ 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 $k I$ 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 $(0, 2)$ .
$(0, 2)$ .

Checking saved progress…

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.

First action, rightmost factor

When do maps commute?

See the method in action.

Reflect then rotate

Write each standard matrix.

Put the first action on the right.

Calculate the combined product.

Identify its geometric action.

Reverse the order

Reflect first, then rotate.

Rotate first, then reflect.

Compare the final points.

Try it yourself.

Check your understanding

Keep practising