A matrix is a rectangular table of numbers. In precalculus and early linear algebra, the useful 2×2 case is a small machine: you feed it a point (or vector), and it outputs a new point. That is how scaling, rotating, and shearing shapes work in graphics and physics demos.
Reading the table
A 2×2 matrix has four entries:
What “apply a matrix” means
For a vector ⟨x, y⟩, the 2×2 matrix produces:
⟨ a·x + b·y , c·x + d·y ⟩
Example: the scaling matrix [[2, 0], [0, 2]] sends ⟨3, 1⟩ to ⟨6, 2⟩ — everything doubles. That is a uniform zoom.
Determinant: how areas change
For [[a, b], [c, d]] the determinant is:
det = a·d − b·c
Its absolute value tells you the area scale factor of the transformation. If |det| = 2, areas of shapes generally double. If det = 0, the matrix squashes everything onto a lower-dimensional set (often a line) — information is lost; the matrix is called singular.
Identity matrix [[1,0],[0,1]] leaves every point alone. Its determinant is 1: areas stay the same, and nothing is flipped.
Common trap: Order of multiplication matters for matrices. “Do scale then rotate” is not always the same as “do rotate then scale.” Treat matrices like operations in sequence, not like ordinary numbers that commute.
Practice in the lab
AP → Matrices: reading entries, determinants, and simple scaling.
Open Hyper-Scale Lab