Core Pure · Linear Transformations · Year 1

Successive Transformations

Combining transformations means combining matrices — but the order you multiply in is the order the transformations are applied, read right to left.

15 min read 4 worked examples

Combining transformations

Theorem 27.1 — composition of transformations

If transformation $M_1$ is applied first, followed by $M_2$, the single matrix representing the combined transformation is $M_2M_1$ — not $M_1M_2$. The matrix applied first is written on the right, closest to the coordinate vector, since $M_2(M_1\mathbf{x})=(M_2M_1)\mathbf{x}$.

Exam technique

Read composite transformations right to left: in $M_2M_1\mathbf{x}$, $\mathbf{x}$ is transformed by $M_1$ first, then the result is transformed by $M_2$. Getting this order backwards is one of the most common errors in this topic.

Example 1 · Combining a rotation and reflection

$M_1=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ represents a $90°$ rotation, applied first, followed by $M_2=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$, a reflection in the $x$-axis. Find the single matrix representing the combined transformation, and describe it.

The combined matrix is $M_2M_1$:

$$M_2M_1=\begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}0&-1\\1&0\end{pmatrix}=\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$$

This composite matrix represents reflection in the line $y=-x$ (matching the standard reflection matrix from the previous lesson).

Example 2 · Finding the image via a composite transformation

Using the same $M_1,M_2$ as Example 1, find the image of $(4,1)$ under "rotation then reflection".

Using the combined matrix $M_2M_1=\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$:

$\begin{pmatrix}0&-1\\-1&0\end{pmatrix}\begin{pmatrix}4\\1\end{pmatrix}=\begin{pmatrix}-1\\-4\end{pmatrix}$.

Order matters: a worked comparison

Example 3 · Showing the two orders give different results

Using $M_1=\begin{pmatrix}2&0\\0&1\end{pmatrix}$ (a stretch) and $M_2=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ (reflection in $y=x$), find both $M_2M_1$ and $M_1M_2$, and confirm they differ.

$M_2M_1=\begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}2&0\\0&1\end{pmatrix}=\begin{pmatrix}0&1\\2&0\end{pmatrix}$.

$M_1M_2=\begin{pmatrix}2&0\\0&1\end{pmatrix}\begin{pmatrix}0&1\\1&0\end{pmatrix}=\begin{pmatrix}0&2\\1&0\end{pmatrix}$.

These are different matrices, confirming $M_2M_1\neq M_1M_2$: "stretch then reflect" is a genuinely different transformation from "reflect then stretch."

Example 4 · Three successive transformations

Find the single matrix representing "rotate $90°$, then enlarge scale factor $3$, then reflect in the $x$-axis," applied in that order.

$R=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, $E=\begin{pmatrix}3&0\\0&3\end{pmatrix}$, $F=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$.

Applied in order $R$, then $E$, then $F$, the combined matrix is $FER$ (last applied on the left):

$ER=\begin{pmatrix}3&0\\0&3\end{pmatrix}\begin{pmatrix}0&-1\\1&0\end{pmatrix}=\begin{pmatrix}0&-3\\3&0\end{pmatrix}$.

$FER=\begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}0&-3\\3&0\end{pmatrix}=\begin{pmatrix}0&-3\\-3&0\end{pmatrix}$.

Common mistake

Writing $M_1M_2$ instead of $M_2M_1$ for "$M_1$ then $M_2$" is the single most common error in this topic. A useful mnemonic: the transformation applied first sits closest to the vector it's multiplying.

Exam-style questions

Question 1 5 marks

$M_1$ represents an enlargement scale factor $2$; $M_2=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ represents reflection in $y=x$. Find the single matrix for "enlargement, then reflection" and use it to find the image of $(3,-1)$.

$M_1=\begin{pmatrix}2&0\\0&2\end{pmatrix}$; combined matrix is $M_2M_1$ (M1)

$M_2M_1=\begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}2&0\\0&2\end{pmatrix}=\begin{pmatrix}0&2\\2&0\end{pmatrix}$ (M1, A1)

$\begin{pmatrix}0&2\\2&0\end{pmatrix}\begin{pmatrix}3\\-1\end{pmatrix}=\begin{pmatrix}-2\\6\end{pmatrix}$ (M1, A1)

Question 2 6 marks

$M_1$ represents rotation by $90°$ anticlockwise, and $M_2$ represents rotation by $180°$. Find the single matrix for "$M_1$ then $M_2$", and identify the overall transformation.

$M_1=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, $M_2=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}$ (B1, B1)

$M_2M_1=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}\begin{pmatrix}0&-1\\1&0\end{pmatrix}=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$ (M1, A1)

This represents rotation by $270°$ ($=90°+180°$) anticlockwise, equivalently $90°$ clockwise. (A1, A1)

Question 3 6 marks

Given $A=\begin{pmatrix}1&2\\0&1\end{pmatrix}$ and $B=\begin{pmatrix}2&0\\1&1\end{pmatrix}$, find $AB$ and $BA$, and use the results to confirm that composing transformations is not commutative.

$AB=\begin{pmatrix}1&2\\0&1\end{pmatrix}\begin{pmatrix}2&0\\1&1\end{pmatrix}=\begin{pmatrix}4&2\\1&1\end{pmatrix}$ (M1, A1)

$BA=\begin{pmatrix}2&0\\1&1\end{pmatrix}\begin{pmatrix}1&2\\0&1\end{pmatrix}=\begin{pmatrix}2&4\\1&3\end{pmatrix}$ (M1, A1)

$AB\neq BA$, confirming that "apply $B$ then $A$" differs from "apply $A$ then $B$." (B1, B1)

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