Introduction

When we apply a geometric or algebraic transformation—like a rotation, reflection, scaling, or shear—we often want to apply more than one in a row.
Matrix multiplication gives us a compact way to combine these steps into a single transformation.

This article explains how and why multiple transformations can be composed using matrices, and how to compute and interpret the results.

Why Compose Transformations?

Some reasons we combine transformations:

• To simplify repeated operations into one matrix.
• To understand how complex motions (rotate + scale + reflect) interact.
• To analyze systems where each step depends on the previous one.
• To reduce computational cost in graphics or simulations.

A sequence like:

• scale → rotate → reflect

can be replaced by one matrix that does all three at once.

Applying Transformations Step-by-Step

Suppose you have a vector $v$ and two transformations:

• First apply matrix $A$
• Then apply matrix $B$

The process is:

1. Compute $A v$
2. Then compute $B (A v)$

This is often written as: $$v' = B A v$$ Key points:

• The matrix closest to $v$ acts first.
• The order matters: in general, $AB \neq BA$.
• Each transformation “feeds into” the next.

Matrix Multiplication as Composition

When you multiply matrices, you are really combining transformations.

If:

• $A$ scales a vector
• $B$ rotates a vector

Then $BA$ is the transformation that:

1. scales first
2. rotates second

This is why the order of multiplication is so important.

A helpful mental model

Think of matrices as machines:

• $A$ is a machine that modifies an input vector.
• $B$ is another machine.

Sending $v$ through both machines in order gives: $$v \xrightarrow{A} A v \xrightarrow{B} B(A v)$$ The combined machine is $BA$.

Examples of Composition

Example 1: Scaling then rotating

Let $$A = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$

• $A$ doubles the size of a vector.
• $B$ rotates a vector by $90^\circ$.

The combined transformation is: $$BA = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix}$$

Example 2: Reflection then shear

Let $R$ reflect across the $x$-axis and $S$ shear horizontally.
Then $SR$ means:

1. reflect
2. shear

But $RS$ means:

1. shear
2. reflect

These produce different results—composition is not commutative.

Geometric Interpretation

When composing transformations:

• Rotations add their angles (if centered at the origin).
• Scalings multiply their scale factors.
• Reflections flip orientation; two reflections may produce a rotation.
• Shears distort shapes; combining shears can mimic rotation or scaling.

Composition lets us build complex transformations from simple ones.

Exercises

1. Let
   $A = \begin{pmatrix}2 & 0 \\ 0 & 1\end{pmatrix}$
   and
   $B = \begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix}$.
   Compute $BA$.

   Solution

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

2. Apply $A$ then $B$ to the vector $v = (3,1)$ using the same matrices as above.

   Solution

   First compute $A v$: $$A(3,1) = (6,1)$$ Then apply $B$: $$B(6,1) = (7,1)$$

3. Suppose $R$ is a $90^\circ$ rotation matrix and $S$ is a scaling matrix with factor $3$.
   Write the combined transformation matrix $RS$.

   Solution

   A $90^\circ$ rotation is: $$R = \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}$$ Scaling by $3$ is: $$S = \begin{pmatrix}3 & 0 \\ 0 & 3\end{pmatrix}$$ Thus: $$RS = \begin{pmatrix} 0 & -3 \\ 3 & 0 \end{pmatrix}$$

4. True or false: If $A$ and $B$ are both reflections, then $AB$ is always a reflection.

   Solution

   False.
   Two reflections can produce a rotation (for example, reflecting across two lines that meet at an angle).

5. Let
   $C = \begin{pmatrix}0 & -1 \\ 1 & 0\end{pmatrix}$
   and
   $D = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$.
   Compute $DC$.

   Solution

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

6. Describe in words what it means for $BA$ to represent “first apply $A$, then apply $B$.”

   Solution

   $BA$ means:

   • take a vector
   • transform it with $A$
   • then transform the result with $B$

   The matrix closest to the vector acts first.

7. Compute the composition $$\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}$$

   Solution

   $$\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 0 & 3 \end{pmatrix}$$