Matrix Transformations (AI HL)
Every reflection, rotation, stretch and enlargement about the origin can be written as a \(2\times2\) matrix, and applying that matrix to a point's coordinates gives the transformed point directly. This page covers how to build and use those matrices for a single transformation, with worked examples and the setup mistakes that lose easy marks. It's part of the broader Matrix Transformations topic.
13 questions on this sub-topic.
Building and applying the matrix
Covered under IB syllabus reference AHL3.9: geometric transformations of points using matrices of the form \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}+\begin{pmatrix}e\\f\end{pmatrix}\). None of the rules below are in the formula booklet - they're definitions and a shortcut for building the matrix.
Applying the matrix to a point
\(\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}ax+by\\cx+dy\end{pmatrix}\)
Write the point as a column vector and multiply on the left by the matrix - the result is the coordinates of the image.
Building the matrix from basis images
Image of \((1,0)\) is column 1; image of \((0,1)\) is column 2
If a question tells you where \(T\) sends \((1,0)\) and \((0,1)\), stack those images as the columns of \(M\) - no multiplying required.
Repeated transformations
Applying the same transformation \(n\) times is the matrix power \(M^n\)
Useful for questions like "describe \(R^2\)" for a rotation matrix \(R\) - raise the matrix to a power rather than composing it with itself step by step.
Ready to combine more than one of these? See Composite Transformations, or the full syllabus table and GDC steps at Matrix Transformations.
Worked examples
Consider transformations about the origin.
(a) Write down the \(2\times2\) matrix for reflection in the \(x\)-axis.
(b) Write down the \(2\times2\) matrix for rotation \(180^\circ\).
(c) Write down the \(2\times2\) matrix for reflection in the line \(y = -x.\)
Worked solution
(a) \(\begin{pmatrix}1&0\\0&-1\end{pmatrix}.\) A1
(b) \(\begin{pmatrix}-1&0\\0&-1\end{pmatrix}.\) A1
(c) \(\begin{pmatrix}0&-1\\-1&0\end{pmatrix}.\) A1
A transformation maps \((1,0)\) to \((0,1)\) and \((0,1)\) to \((-1,0).\) Find the transformation matrix and describe it.
(a) State the top-left entry.
(b) State the top-right entry.
(c) State the bottom-left entry.
(d) State the bottom-right entry.
(e) Describe the transformation.
Worked solution
The images of \((1,0)\) and \((0,1)\) form the columns of the matrix. M1 \[M=\begin{pmatrix}0&-1\\1&0\end{pmatrix}.\] A1 A1 A1 This is a \(90^\circ\) anticlockwise rotation. A1
Common mistakes
- Writing the point as a row vector instead of a column vector. Always multiply the transformation matrix by the point written as a column vector, \(M\begin{pmatrix}x\\y\end{pmatrix}\) - multiplying the other way round gives a different, incorrect result.
- Swapping the columns when building a matrix from basis images. The image of \((1,0)\) is always column 1 and the image of \((0,1)\) is always column 2 - putting them in the wrong order produces the transpose of the correct matrix.
- Assuming a shear changes area. Every shear matrix has determinant 1, so shears always preserve area - only the shape distorts, never the size.
Ready to practise properly?
13 matrix-transformation questions, marked instantly like the real exam.
Quick answers
How do you apply a transformation matrix to a point?
Write the point as a column vector and multiply it on the left by the transformation matrix: \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}ax+by\\cx+dy\end{pmatrix}.\)
How do you find the matrix of a transformation from where it sends the basis vectors?
The image of \((1,0)\) becomes the first column of the matrix and the image of \((0,1)\) becomes the second column.