How to Combine Transformation Matrices

Transformation matrices are combined by composition: apply one transformation to a point, then apply the next transformation to the result. The main issue is the order of multiplication.

The convention used here is the standard computer-graphics and linear-algebra convention of column vectors. Under this convention, if $A$ is applied first and $B$ second, the combined transformation is $BA$, not $AB$.

The basic rule

Let a point or vector be represented by a column vector $\mathbf{x}$. If the first transformation is represented by $A$ and the second by $B$, then

$$\mathbf{x}' = A\mathbf{x}$$

and then

$$\mathbf{x}'' = B\mathbf{x}' = B(A\mathbf{x}) = (BA)\mathbf{x}.$$

Therefore, the combined matrix is

$$M = BA.$$

Read the product from right to left: the rightmost matrix acts first.

For several transformations, the same principle gives

$$M = A_n A_{n-1}\cdots A_2 A_1,$$

where $A_1$ is applied first and $A_n$ is applied last.

Combining linear transformations

In two dimensions, a linear transformation has the form

$$\begin{bmatrix}x'\\y'\end{bmatrix} = A\begin{bmatrix}x\\y\end{bmatrix},$$

where $A$ is a $2\times 2$ matrix. For example, a counterclockwise rotation by angle $\theta$ is

$$R(\theta)= \begin{bmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{bmatrix}.$$

Scaling by factors $s_x$ and $s_y$ is represented by

$$S= \begin{bmatrix} s_x & 0\\ 0 & s_y \end{bmatrix}.$$

If you rotate first and scale second, the combined matrix is

$$M=SR(\theta).$$

If you scale first and rotate second, it is instead

$$M=R(\theta)S.$$

These products are generally different because matrix multiplication is not commutative:

$$SR(\theta)\ne R(\theta)S.$$

Including translations with homogeneous coordinates

A translation cannot be represented by an ordinary $2\times 2$ matrix acting on $(x,y)$, because translation is not a linear transformation. Introduce a third coordinate and write the point as

$$\mathbf{p}=\begin{bmatrix}x\\y\\1\end{bmatrix}.$$

A translation by $(t_x,t_y)$ is then represented by the $3\times 3$ matrix

$$T= \begin{bmatrix} 1&0&t_x\\ 0&1&t_y\\ 0&0&1 \end{bmatrix}.$$

For example, the homogeneous matrices for scaling and rotation are

$$S= \begin{bmatrix} s_x&0&0\\ 0&s_y&0\\ 0&0&1 \end{bmatrix}, \qquad R(\theta)= \begin{bmatrix} \cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\ 0&0&1 \end{bmatrix}.$$

This lets you combine scaling, rotation, reflection, shear, and translation in one product.

Fully worked example

Suppose a point is transformed in this order:

1. Rotate it counterclockwise by $90^\circ$.
2. Scale the $x$-coordinate by $2$ and the $y$-coordinate by $3$.
3. Translate it by $(1,-2)$.

Using homogeneous coordinates, the matrices are

$$R= \begin{bmatrix} 0&-1&0\\ 1&0&0\\ 0&0&1 \end{bmatrix}, \qquad S= \begin{bmatrix} 2&0&0\\ 0&3&0\\ 0&0&1 \end{bmatrix}, \qquad T= \begin{bmatrix} 1&0&1\\ 0&1&-2\\ 0&0&1 \end{bmatrix}.$$

Because the order is rotation, then scaling, then translation, the combined matrix is

$$M=TSR.$$

First multiply $S$ and $R$:

$$SR= \begin{bmatrix} 0&-2&0\\ 3&0&0\\ 0&0&1 \end{bmatrix}.$$

Now multiply by $T$:

$$M=TSR= \begin{bmatrix} 0&-2&1\\ 3&0&-2\\ 0&0&1 \end{bmatrix}.$$

Thus every point can be transformed in one step using

$$\begin{bmatrix} x'\\y'\\1 \end{bmatrix} = \begin{bmatrix} 0&-2&1\\ 3&0&-2\\ 0&0&1 \end{bmatrix} \begin{bmatrix} x\\y\\1 \end{bmatrix}.$$

For the point $(1,2)$,

$$M\begin{bmatrix}1\\2\\1\end{bmatrix} = \begin{bmatrix}-3\\1\\1\end{bmatrix}.$$

Checking step by step gives the same result: rotation sends $(1,2)$ to $(-2,1)$, scaling sends this to $(-4,3)$, and translation sends it to $(-3,1)$.

Common mistakes

Reversing the order

For column vectors, the first operation is the rightmost factor. If rotation comes first and translation comes second, use $TR$, not $RT$.

Multiplying entries element by element

Combining matrices means ordinary matrix multiplication. It does not mean multiplying corresponding entries. Each entry of a product is obtained from a row-column dot product.

Treating translations as $2\times 2$ matrices

A translation requires homogeneous coordinates, usually a $3\times 3$ matrix in two dimensions and a $4\times 4$ matrix in three dimensions.

Mixing row and column conventions

Some applications use row vectors, written as $\mathbf{x}^{\mathsf{T}}A$. In that convention, the multiplication order is reversed: if $A$ is applied first and $B$ second, the combined matrix is $AB$. Neither convention is wrong, but the vector orientation and order must remain consistent.

Ignoring coordinate systems

A transformation may be described in world coordinates, object coordinates, or coordinates relative to a pivot. Transforming about a pivot often requires several matrices. For example, to rotate about a point $\mathbf{c}$, translate by $-\mathbf{c}$, rotate, then translate by $\mathbf{c}$:

$$M=T_{\mathbf{c}}RT_{-\mathbf{c}}.$$

Again, the rightmost operation happens first.

A practical verification method

After forming the product, test it on a simple point such as $(1,0)$ or $(0,0)$. Compute the result both ways: apply each transformation separately, and apply the combined matrix once. If the answers differ, check the multiplication order, vector convention, and matrix dimensions.