Mathematics ★☆☆ Easy

⬜ Matrix Transformations

Edit the 2×2 matrix entries and watch the coordinate grid transform in real time. Rotate, scale, shear, reflect — see eigenvectors glow as they stay on their lines, and watch the determinant measure the area change.

Presets:
1.00
0.00
0.00
1.00
M = ⎡ 1.00 0.00 ⎤ ⎣ 0.00 1.00 ⎦
det(M) = 1.00  |  Area scale: 1.00×
Trace: 2.00 Det: 1.00 λ₁: 1.00 λ₂: 1.00 Type: Identity

How to read this

The light grey grid shows the original coordinate system. The coloured grid shows the same grid after the matrix transformation. The red arrow is the transformed î = (1,0) basis vector; the blue arrow is ĵ = (0,1). Eigenvectors (yellow) are special vectors that only stretch or flip — they never rotate.

Key concepts

A 2×2 matrix M = [[a,b],[c,d]] transforms every point (x,y) → (ax+by, cx+dy). The determinant det(M) = ad−bc measures how area changes: det=1 preserves area, det=−1 reflects, det=0 collapses space to a line (or point).

Eigenvalues λ satisfy Mv = λv — the matrix only stretches the eigenvector by λ, never rotates it. Real eigenvalues exist when the discriminant (trace² − 4·det) ≥ 0. λ = (trace ± √(trace²−4·det)) / 2.