1 · IB Maths
Matrix transformations
A 2×2 matrix moves every point in the plane. Set its four entries, watch the unit square and the grid move, and read off the determinant as the area scale factor.
Predict first
Start from the default matrix (a = 1.2, b = −0.6, c = 0.4, d = 1.0). Set b to 0 and leave a, c and d alone. What happens to the determinant?
det = ad − bc. With b = −0.6 it is 1.2×1.0 − (−0.6)(0.4) = 1.44. With b = 0 it is 1.2×1.0 − 0 = 1.2, which is smaller.
Check it in the model: Move the b slider from −0.60 to 0.00 and read the Determinant (area factor) readout (1.44 then 1.2).
image of î (first column) image of ĵ (second column) transformed grid
x′ = a x + b y
y′ = c x + d y
det M = ad − bc
λ² − (a + d)λ + det M = 0
Try this
- Set a = 2, d = 2, b = c = 0. Predict the determinant and the area of the image of the unit square, then check both readouts.
- Use the Rotation 45° preset and then the Shear preset. Which has a determinant of 1, and what does that say about area?
- Start from the Collapse preset and change d from 1 to 1.5. What happens to the determinant and the inverse?
Exam tip: read the columns of the matrix as the images of the basis vectors (1, 0) and (0, 1); a determinant of 0 means the matrix has no inverse, and a negative determinant means orientation is reversed.
More on 1: Arithmetic and geometric series · Compound interest and depreciation · Proof by induction · Permutations and combinations · De Moivre's theorem · Systems of linear equations · Complex numbers on the Argand plane · Roots of a complex number · Complex loci · Sequences and series · Exponentials and logarithms · Binomial theorem