How Do You Use a Matrix to Represent a Reflection, Stretch, Rotation, or Translation?

A transformation matrix maps a point, written as a column vector, to its image. Reflections, stretches, and rotations use $\mathbf{x}'=A\mathbf{x}$, but translation requires $\mathbf{x}'=A\mathbf{x}+\mathbf{b}$, because an ordinary $2\times2$ matrix always leaves the origin fixed.

This is HL-only content from AHL 3.9, Matrix transformations, and may be tested on any GDC-active HL paper.

For $\mathbf{x}=\begin{pmatrix}x\\y\end{pmatrix}$, multiply the matrix by the vector:

$$\begin{pmatrix}x'\\y'\end{pmatrix}=A\begin{pmatrix}x\\y\end{pmatrix}.$$

Matrix multiplication works because each row determines one new coordinate. For example, reflecting in the $x$-axis keeps the $x$-coordinate unchanged and multiplies the $y$-coordinate by $-1$. A rotation matrix combines both original coordinates using sine and cosine.

For example, rotate $(2,1)$ anticlockwise through $90^\circ$, then translate by $\begin{pmatrix}3\\-2\end{pmatrix}$:

$$\left(\begin{array}{c}{x}^{\mathrm{\prime }}\\ y^{\mathrm{\prime }}\end{array}\right)=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\begin{array}{c}2\\ 1\end{array}\right)+\left(\begin{array}{c}3\\ -2\end{array}\right)=\left(\begin{array}{c}-1\\ 2\end{array}\right)+\left(\begin{array}{c}3\\ -2\end{array}\right)=$$

The rotation sends $(2,1)$ to $(-1,2)$; the translation then adds $3$ horizontally and $-2$ vertically.

For successive linear transformations, order matters. If $A$ acts first and $B$ acts second, then $\mathbf{x}'=B(A\mathbf{x})=(BA)\mathbf{x}$. Matrix multiplication is , so generally .

To transform an entire shape, apply the same rule to every vertex and join the image points in the corresponding order. The determinant provides a useful check: a negative determinant reverses orientation, while its absolute value gives the area scale factor. For example, a horizontal stretch by $2$ followed by a vertical stretch by $3$ has determinant $6$, so every area is multiplied by $6$. A rotation has determinant $1$, so it preserves area. A reflection has determinant $-1$, preserving area but reversing orientation. These checks can reveal sign errors.

A common misconception is that translation can be represented by an ordinary $2\times2$ matrix. It cannot; the translation vector must be added separately after the matrix multiplication.

Exam technique: preserve the stated order, show the matrix-vector setup, and give the final transformed coordinates. If asked about area, remember that the area scale factor is $|\det A|$.