Image = matrix × column vector: Picture a robotic arm picking up a dot on graph paper and dropping it somewhere new. A 2×2 matrix is the instruction for that move — and it moves every point the same way.
Write the point as a column vector and multiply on the left:
image .
The order matters — the matrix goes on the left, the point on the right.
IB-style question — image of a point
A design app stores a logo pixel at the point P(3, 1). The transformation is applied.
Find the image of P.
Step by step
- Write P as a column vector and multiply on the left.
- Top entry: row 1 times the column = 2·3 + 0·1.
- Bottom entry: row 2 times the column = 1·3 + 3·1.
Final answer
The image is P′(6, 6). (On a GDC: enter the 2×2 matrix and the 2×1 vector and multiply.)
Why a column on the right?: Each new coordinate is a mix of the old x and y. The top row of the matrix builds the new x; the bottom row builds the new y. Keeping points as columns lets one matrix transform many points at once (stack them as columns of a bigger matrix).
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Rotation, reflection, enlargement — about the origin: Three transformations have set 2×2 matrices (all about the origin, which stays fixed):
Rotation anticlockwise by angle : .
Enlargement scale factor : — multiplies every distance from O by .
Reflection in the line : .
A stretch uses different scale factors on each axis: .
IB-style question — write down a rotation matrix
A wind-turbine blade is modelled by the point B(4, 0). It rotates 90° anticlockwise about the hub at the origin.
Find the matrix and the image of B.
Step by step
- Use the rotation matrix with θ = 90°, so cos 90° = 0, sin 90° = 1.
- Apply it to B = (4, 0).
Final answer
Matrix ; the blade tip moves to (0, 4) — straight up, as a 90° turn should.
IB-style question — enlargement
A photo-editing tool enlarges an image about the origin by scale factor 3.
Write down the transformation matrix and find the image of the corner (2, −1).
Step by step
- Enlargement scale factor 3 has k on the diagonal.
- Apply it to (2, −1) — every coordinate is tripled.
Final answer
Matrix ; the corner moves to (6, −3).
Recognise a matrix in reverse: You'll also be asked to describe the transformation a matrix represents. Check the shape: a diagonal is an enlargement; matches ; flips the sign of y, so it's a reflection in the x-axis.