Matrix Transformations (AI HL)
A 2 by 2 matrix can rotate, reflect, stretch or enlarge every point in the plane in one multiplication - which is why matrices are the natural language for describing geometric transformations. This topic covers writing down the standard transformation matrices, applying them to points, combining several transformations into a single matrix, and using the determinant to read off the area scale factor and orientation of a transformation.
What the syllabus says
This topic maps onto a single point in the official IB Applications & Interpretation syllabus, examinable at HL only.
| Code | Syllabus content |
|---|---|
| AHL3.9 | Geometric transformations of points in two dimensions using matrices: reflections, horizontal and vertical stretches, enlargements, translations and rotations. Matrix transformations of the form \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}+\begin{pmatrix}e\\f\end{pmatrix}\). Compositions of the above transformations. Geometric interpretation of the determinant of a transformation matrix: area of image \(=|\det A|\times\)area of object. |
Rotations, reflections, stretches and enlargements about the origin are pure 2 by 2 matrix multiplications; a translation needs the extra \((e,f)\) vector added on, since it can't be written as a matrix multiplication alone.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is a transformation matrix?
A transformation matrix is a 2 by 2 matrix that, when multiplied by a point's column vector, produces the image of that point under a geometric transformation. Multiplying by the same matrix transforms every point in the plane the same way.
e.g. the reflection matrix \(\begin{pmatrix}1&0\\0&-1\end{pmatrix}\) sends \((3,2)\) to \((3,-2)\), since \(\begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}3\\2\end{pmatrix}=\begin{pmatrix}3\\-2\end{pmatrix}.\)
What is a rotation matrix?
A rotation matrix turns every point about the origin through a fixed angle. For an anticlockwise rotation through angle \(\theta\), the matrix is \(\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}.\)
e.g. rotating \((1,0)\) by \(90^\circ\) anticlockwise: \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}=\begin{pmatrix}0\\1\end{pmatrix}.\)
What does the determinant tell you?
The determinant of a transformation matrix gives its area scale factor - the absolute value tells you how much areas are stretched or shrunk, and a negative sign means orientation is reversed.
e.g. for \(\begin{pmatrix}3&0\\0&2\end{pmatrix}\), \(\det = 3(2)-0(0)=6\), so every area is scaled by a factor of 6.
What is a composition of transformations?
A composition applies two or more transformations one after another, and can always be written as a single matrix formed by multiplying the individual matrices together - in the order that puts the first transformation applied on the right.
e.g. rotating \((1,0)\) by \(90^\circ\) gives \((0,1)\); reflecting that result in the \(x\)-axis then gives \((0,-1)\).
What is an invariant point?
An invariant point is a point that maps to itself under a transformation - it doesn't move. Every transformation matrix fixes the origin, and some (like shears) fix an entire line of points.
e.g. the shear \(\begin{pmatrix}1&0\\3&1\end{pmatrix}\) leaves \((0,2)\) unchanged, since \(\begin{pmatrix}1&0\\3&1\end{pmatrix}\begin{pmatrix}0\\2\end{pmatrix}=\begin{pmatrix}0\\2\end{pmatrix}.\)
Key formulas
A handful of standard matrices cover every transformation on this topic - the tables below summarise them, and the cards after go into more depth on each one.
Formula reference
The rotation matrix and the area scale factor result are both in the formula booklet; applying a general matrix and composing transformations are treated as definitions rather than booklet formulas.
| Formula | Used for | Booklet? |
|---|---|---|
| \(\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}ax+by\\cx+dy\end{pmatrix}\) | Applying a transformation matrix to a point | Not in booklet - definition |
| \(R=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}\) | Anticlockwise rotation about the origin | ✓ Yes |
| Area of image \(=|\det A|\times\)area of object | Area scale factor of a transformation | ✓ Yes |
| \(T = S\,R\) applies \(R\) first, then \(S\) | Composing two transformations into one matrix | Not in booklet - procedure |
Positive vs negative determinant
The sign of the determinant tells you whether orientation is preserved or reversed - its size always gives the area scale factor.
| Feature | Positive determinant | Negative determinant |
|---|---|---|
| Orientation | Preserved - shapes keep the same "handedness" | Reversed - a reflection is involved |
| Area scale factor | \(\det A\) | \(|\det A|\) (always positive) |
| Typical example | A rotation or an enlargement | A reflection, or a composition including one |
Standard transformation matrices
Each of these is a pure 2 by 2 matrix applied about the origin - a translation needs a vector added on separately, since it isn't linear.
Rotation
\[\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}\]
Rotates every point anticlockwise about the origin through angle \(\theta\).
✓ In the formula bookletReflection
\[\begin{pmatrix}0&1\\1&0\end{pmatrix} \text{ reflects in } y=x\]
Common reflection matrices swap or negate coordinates - in \(y=x\), \((x,y)\to(y,x)\); in the \(x\)-axis, \((x,y)\to(x,-y)\).
Not in the formula booklet - standard cases to knowStretch or enlargement
\[\begin{pmatrix}k&0\\0&k\end{pmatrix} \text{ enlarges by scale factor } k\]
A stretch scales in one direction only (one diagonal entry \(\neq1\)); an enlargement scales both directions by the same factor.
Not in the formula booklet - standard cases to knowDeterminants and area
The determinant links the algebra of a matrix directly to the geometry of the shape it transforms.
Area scale factor
The area of any shape's image is \(|\det A|\) times the area of the original - this holds for every transformation matrix, not just special cases.
✓ In the formula bookletSign of the determinant
A positive determinant preserves orientation; a negative one reverses it, which always signals that a reflection is part of the transformation.
Not in the formula booklet - interpretationInvertibility
A transformation matrix is invertible exactly when \(\det A \neq 0\) - if the determinant is zero, the transformation collapses the plane and can't be undone.
Not in the formula booklet - consequence of the determinantComposing transformations
Several transformations chained together always reduce to a single matrix - but only if you multiply them in the right order.
Order matters
"Rotation \(R\) followed by enlargement \(S\)" is the single matrix \(SR\), not \(RS\) - the transformation applied first sits on the right of the product.
Not in the formula booklet - matrix multiplication is not commutativeRepeated transformations
Applying the same transformation \(n\) times is the matrix power \(M^n\) - useful for questions like "describe \(R^2\)" for a rotation matrix \(R\).
Not in the formula booklet - repeated compositionInverse transformation
To find the point that maps to a given image, apply \(M^{-1}\) to the image vector - this only works when \(\det M \neq 0\).
Not in the formula booklet - consequence of invertibilityWorked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
The matrix \(M=\begin{pmatrix}0&1\\1&0\end{pmatrix}\) reflects in the line \(y=x:\)
(a)(i) Find the image of \((5,-2).\)
(a)(ii) Find the image of \((-3,4).\)
(b) Find the image of \((0,0).\)
Worked solution
(a)(i) Coordinates swap. M1
\((5,-2)\to(-2,5)\); A1
(a)(ii) \((-3,4)\to(4,-3).\) A1
(b) \((0,0).\) A1
The matrix \(R=\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) represents a \(90^\circ\) anticlockwise rotation and \(S=\begin{pmatrix}2&0\\0&2\end{pmatrix}\) an enlargement.
(a) Find the single matrix \(T=SR\) representing the rotation followed by the enlargement.
(b) Find the image of the point \((3,1)\) under \(T.\)
(c) Find the area scale factor of \(T.\)
Worked solution
(a) \(T=SR=\begin{pmatrix}2&0\\0&2\end{pmatrix}\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) M1
\(=\begin{pmatrix}0&-2\\2&0\end{pmatrix}.\) A1
(b) \(T\begin{pmatrix}3\\1\end{pmatrix}\) M1
\(=\begin{pmatrix}-2\\6\end{pmatrix},\) image \((-2,6).\) A1
(c) Area scale factor \(=|\det T|=|0-(-4)|\) M1
\(=4.\) A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Multiplying composed transformations in the wrong order. "Rotation \(R\) followed by enlargement \(S\)" is the matrix \(SR\), with \(S\) on the left - not \(RS\). Matrix multiplication is not commutative, so getting the order backwards gives the wrong combined transformation.
- Forgetting the area scale factor uses \(|\det A|\), not \(\det A\). Area can never be negative, even though a determinant can be - always take the absolute value before stating an area scale factor.
- Assuming a shear changes area. Every shear matrix has determinant 1, so shears always preserve area - only the shape distorts, never the size.
- 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.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
Inverses, determinants and matrix multiplication are quick and reliable on the GDC - essential for composing transformations, finding area scale factors, and locating pre-images with the inverse.
- Enter each transformation matrix into the calculator's matrix editor before combining or evaluating them.
- Enter a matrix with 2nd → MATRX → EDIT. Determinant: det([A]); inverse: [A] then the x⁻¹ key.TI-84
- Use the matrix template (or menu → Matrix & Vector). det(), and ^-1 for the inverse.Nspire
- Run-Matrix → MAT/VCT (F3) to enter a matrix; use Det and the x⁻¹ key.Casio
- Solve \(Ax=b\) (e.g. finding a pre-image) by computing \([A]^{-1}[b]\).
Tip: If \(\det(A) = 0\), the matrix has no inverse - the transformation collapses the plane and can't be undone.
Questions that ask you to describe \(R^2\) or apply a transformation several times in a row are matrix powers - the GDC raises the whole matrix to a power in one step instead of multiplying it out by hand.
- Enter the transformation matrix and the point (or state) it acts on.
- Enter [A] in 2nd → MATRX → EDIT, then compute [A]^n × [B] on the home screen.TI-84
- Enter the matrix, then type matrix ^ n × the point vector.Nspire
- Run-Matrix → MAT to enter the matrix; compute Mat A ^ n × the point vector.Casio
- For a rotation matrix, \(R^n\) is itself a rotation through \(n\) times the original angle - check your answer makes geometric sense.
Tip: Set the angle mode (degrees or radians) correctly before evaluating a rotation matrix numerically - a mode mismatch silently gives the wrong entries.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Matrix transformation questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
Does the order I multiply transformation matrices in matter?
Yes. For a rotation R followed by an enlargement S, the combined matrix is SR, not RS - the transformation applied first goes on the right. Matrix multiplication is not commutative, so swapping the order generally gives a different combined transformation.
What does the determinant of a transformation matrix tell you?
Its absolute value is the area scale factor - how much the transformation stretches or shrinks area. A negative determinant means the transformation also reverses orientation, which happens whenever a reflection is involved.
How do I find the point that maps to a given image?
Apply the inverse matrix to the image point. If M is the transformation matrix and the image is the vector v, the original point is M-1v, provided det(M) is not zero.
Can translations be written as a 2 by 2 matrix?
No - a translation shifts every point by a fixed vector rather than scaling or rotating around the origin, so it can't be represented by multiplying by a 2 by 2 matrix alone. Rotations, reflections, stretches and enlargements about the origin can all be written this way. See the GDC guide for more calculator-specific instructions.
Sub-topics
Matrix Transformations broken down into its individual skills, each with its own focused page.
Related topics
More Geometry & Trigonometry topics from the same AI HL syllabus unit, in case you want to keep going.