How Do You Find the Composition of Multiple Transformation Matrices?

To compose multiple transformation matrices, multiply them in the reverse order from which the transformations are applied. If transformation $A$ occurs first, followed by $B$ and then $C$, the composite matrix is $CBA$.

This is AHL 3.9: Matrix transformations, so it is HL-only content. The explanation assumes an IB Mathematics: Applications and Interpretation HL short-response setting, although the method can appear on any GDC-active HL paper.

Why Is the Order Reversed?

IB Mathematics represents each point by a column vector $\mathbf{x}$. Applying the transformations successively gives

$$\mathbf{x}\mapsto A\mathbf{x}\mapsto B(A\mathbf{x})\mapsto C(B(A\mathbf{x}))=CBA\mathbf{x}.$$

The matrix nearest the vector acts first; each later matrix is placed to its left. This is function composition written in matrix form.

Matrix multiplication is associative, so brackets may be regrouped without changing the result. However, it is generally not commutative: $AB\neq BA$. Reversing two matrices can therefore change the image, orientation, scale, or position of a shape.

Worked Example

Let $A$ reflect points in the $x$-axis, $B$ rotate them $90^\circ$ anticlockwise, and $C$ stretch horizontally by scale factor $2$:

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

Because the operations occur as $A$, then $B$, then $C$, calculate $CBA$, not $ABC$:

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

For $\mathbf{x}=\begin{pmatrix}3\\1\end{pmatrix}$, the image is

$$CBA\mathbf{x}=\begin{pmatrix}2\\3\end{pmatrix}.$$

Reliable Procedure

List the transformations in the order they occur, then write the image chain from the original column vector. Read the matrices from right to left, multiply them, and apply the result to the vector. If only the composite matrix is requested, stop there.

Check the result by applying each matrix separately to the same starting point. The final coordinates must match the single composite matrix. This is useful because an incorrect order can still produce plausible-looking numbers.

Exam Technique

The common misconception is multiplying in the stated order as $ABC$, which makes the first transformation act last. In an IB response, show the ordered product before using the GDC and state the image as a column vector. This visible set-up provides method evidence.