IB Maths AI HL Matrix Transformations Paper 1 & 2 Rotations & reflections ~9 min read

Matrices of Geometric Transformations

Each geometric transformation — rotation, reflection, enlargement, stretch — has its own 2×2 matrix. Most are given in the formula booklet, but you can always derive any matrix yourself by tracking where the basis vectors (1, 0) and (0, 1) get sent. Those two images become the columns of T. If the transformation also moves the origin, the extra shift is the translation vector (e, f).

📘 What you need to know

Columns of T = images of basis vectors: if (1, 0) maps to (a, c) and (0, 1) maps to (b, d), then T = (abcd).
When there’s a translation: image of (0, 0) gives the translation vector (e, f). Subtract this from the other two images before reading off the columns of T.
Three pairs of points are needed to determine a full matrix transformation T + (e, f) — substitute each pair into the equation and solve.
Rotation anticlockwise by angle θ about origin: T = (cos θ−sin θsin θcos θ). Clockwise has signs of sin θ flipped.
Reflection in the line y = (tan θ)x: T = (cos 2θsin 2θsin 2θ−cos 2θ). For y = mx, solve tan θ = m first.
Enlargement scale factor k about origin: T = (k00k). Horizontal stretch uses (k, 0; 0, 1); vertical stretch uses (1, 0; 0, k).
Translation by (p, q): no matrix needed — just add the column vector (p, q) after T. The translation matrix is not in the formula booklet.

Building the matrix from the images of (1, 0) and (0, 1)

This is the cleanest way to find any transformation matrix. The point (1, 0) ends up at whatever the first column of T says; the point (0, 1) ends up at whatever the second column says. So if you know where these two unit vectors get sent, you can read T off directly. If the origin moves too, the image of (0, 0) gives the translation vector — subtract that off the other images before extracting the columns.

Under a 90° anticlockwise rotation about the origin, (1, 0) → (0, 1) and (0, 1) → (−1, 0). These images are the columns of T = (0−110) — first column is image of e₁, second is image of e₂.

Matrix from basis vector images T = (abcd) where (1, 0) → (a, c) and (0, 1) → (b, d) with translation: (0, 0) → (e, f) — subtract from other images before extracting T

Standard transformation matrices

Most of the matrices below are in the formula booklet; the translation is the only one you need to remember separately, since it’s added as a column vector rather than multiplied. For a reflection in y = mx, you first convert m to an angle by solving tan θ = m, then plug θ into the reflection formula. Notice the double-angle: the reflection matrix uses 2θ, while rotations use just θ.

Transformation	Matrix T (apply to (x, y)^T)
Rotation anticlockwise by angle θ about origin	(cos θ−sin θsin θcos θ)
Rotation clockwise by angle θ about origin	(cos θsin θ−sin θcos θ)
Reflection in the line y = (tan θ)x	(cos 2θsin 2θsin 2θ−cos 2θ)
Enlargement, scale factor k, centre origin	(k00k)
Horizontal stretch parallel to x-axis, scale factor k	(k001)
Vertical stretch parallel to y-axis, scale factor k	(100k)
Translation by vector (p, q) — not in booklet	add (pq) after T

Self-check exercise: try deriving each formula yourself by working out where (1, 0) and (0, 1) get sent under the named transformation. For instance, a 90° anticlockwise rotation sends (1, 0) → (0, 1) and (0, 1) → (−1, 0); plug θ = 90° into the rotation formula and check the columns match.

🧭 Recipe — finding the matrix of a transformation

Named transformation (rotation, reflection, enlargement, stretch)? — use the formula booklet entry directly. For reflection in y = mx, find θ from tan θ = m first.
Given object & image points? — check what the origin maps to first.
Origin fixed (no translation) — the images of (1, 0) and (0, 1) become the columns of T directly.
Origin moves — image of (0, 0) = (e, f) is the translation. Subtract it from the other images, then read off T.
No nice points given? — substitute three object/image pairs into the general equation and solve the 6 simultaneous equations for a, b, c, d, e, f.

Worked examples

WE 1

Find T from images of (1, 0) and (0, 1) — no translation

Under a transformation T about the origin (no translation), (1, 0) maps to (3, −1) and (0, 1) maps to (5, 0). Find the matrix T.

images of the basis vectors → columns of T column 1 = image of (1,0) = (3, −1) column 2 = image of (0,1) = (5, 0) stack them as a matrix T = (35−10) no need to solve equations — just read the images off into columns.

WE 2

Find T and the translation vector together

A transformation sends (0, 0) → (2, 5), (1, 0) → (3, −1) and (0, 1) → (5, 0). Find the matrix T and the translation vector (e, f).

(0, 0) → (e, f), so translation (e, f) = (2, 5) subtract translation from other images image of (1,0) without translation: (3−2, −1−5) = (1, −6) image of (0,1) without translation: (5−2, 0−5) = (3, −5) these are now the columns of T T = (13−6−5), (e, f) = (2, 5) image of origin gives the translation — always do that subtraction first.

WE 3

Write the matrix for a 90° anticlockwise rotation

Find the matrix T that represents a rotation of 90° anticlockwise about the origin. Use T to find the image of the point P(4, 1).

use formula with θ = 90° T = (cos 90°−sin 90°sin 90°cos 90°) = (0−110) apply to P(4, 1) top: 0(4) + (−1)(1) = −1 bot: 1(4) + 0(1) = 4 T = (0−110); P′ = (−1, 4) 90° CCW: (x, y) → (−y, x) — quick sanity check.

WE 4

Reflection in y = x√3

Triangle PQR has coordinates P(−1, 4), Q(5, 4), R(2, −1). Find the matrix T that represents a reflection in the line y = x√3, then find the image of P under T.

find θ from tan θ = √3 tan θ = √3 → θ = 60° use formula with 2θ = 120° T = (cos 120°sin 120°sin 120°−cos 120°) = (−½√3/2√3/2½) apply to P(−1, 4) top: −½(−1) + (√3/2)(4) = ½ + 2√3 bot: (√3/2)(−1) + ½(4) = 2 − √3/2 P′ = (½ + 2√3, 2 − √3/2) reflection uses 2θ in the formula — common slip is to use θ.

WE 5

Enlargement and image of a triangle

Write the matrix that represents an enlargement of scale factor 3 about the origin. Use it to find the image of the triangle with vertices A(1, 2), B(4, 0), C(2, 5).

enlargement scale 3, centre origin T = (3003) multiply each coordinate by 3 A → (3, 6) B → (12, 0) C → (6, 15) A′(3, 6), B′(12, 0), C′(6, 15) enlargement about origin: every coordinate scaled by k.

WE 6

Identify the transformation from its matrix

The matrix T = (0110) represents a transformation about the origin. Describe the geometric transformation T fully.

check images of basis vectors T(1,0) = (0, 1) → up the y-axis T(0,1) = (1, 0) → along the x-axis basis vectors swap → reflection in y = x verify with formula: y = (tan θ)x with tan θ = 1 θ = 45°, so 2θ = 90° T = (cos 90°sin 90°sin 90°−cos 90°) = (0110) ✓ reflection in the line y = x reading the columns + comparing to the formula table identifies any standard transformation.

💡 Top tips

Track (1, 0) and (0, 1) first — their images give you the columns of T for free. This works for any non-translation transformation, no equations needed.
Image of the origin = translation — if (0, 0) does not stay at (0, 0), the difference IS the translation vector. Subtract it from every other image before finding T.
Reflection formula uses 2θ, but you find θ from y = (tan θ)x. Easy to confuse with rotation, which uses just θ.
Formula booklet is your friend — all the standard transformation matrices except translation are in there. Don’t memorise; look them up.
Sanity check by trying (1, 0) — for any T you’ve written, multiply T by (10) and check the result matches where (1, 0) should go geometrically.

⚠ Common mistakes

Using θ instead of 2θ in the reflection matrix — the formula booklet entry is cos 2θ, sin 2θ, −cos 2θ. Always double the angle.
Rows mixed up with columns — images of (1, 0) and (0, 1) are the columns of T, not the rows. If you write them as rows, every image will come out wrong.
Forgetting to subtract the translation — if (0, 0) does not map to (0, 0), trying to read T directly from the images of (1, 0) and (0, 1) gives wrong columns.
Mixing up clockwise and anticlockwise rotation matrices — they differ in the sign of the sin θ terms. CCW: minus sign in top-right; CW: minus sign in bottom-left.
Treating enlargement about a non-origin centre as just (k00k) — the booklet formula only works for centre at the origin. Other centres need a translation as well.

Next up — Matrices of Composite Transformations. When two transformations are applied one after another, the combined effect is captured by multiplying the matrices in the right order: ST for “T first, then S”. Order matters — rotation-then-reflection is not the same as reflection-then-rotation.

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