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Mathematics โ˜…โ˜†โ˜† Easy

โฌœ Matrix Transformations

Edit the 2ร—2 matrix entries and watch the coordinate grid transform in real time. Rotate, scale, shear, reflect โ€” 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 = โŽก 1.00 0.00 โŽค โŽฃ 0.00 1.00 โŽฆ
det(M) = 1.00  |  Area scale: 1.00ร—
Trace: 2.00 Det: 1.00 ฮปโ‚: 1.00 ฮปโ‚‚: 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 รฎ = (1,0) basis vector; the blue arrow is ฤต = (0,1). Eigenvectors (yellow) are special vectors that only stretch or flip โ€” they never rotate.

Key concepts

A 2ร—2 matrix M = [[a,b],[c,d]] transforms every point (x,y) โ†’ (ax+by, cx+dy). The determinant det(M) = adโˆ’bc measures how area changes: det=1 preserves area, det=โˆ’1 reflects, det=0 collapses space to a line (or point).

Eigenvalues ฮป satisfy Mv = ฮปv โ€” the matrix only stretches the eigenvector by ฮป, never rotates it. Real eigenvalues exist when the discriminant (traceยฒ โˆ’ 4ยทdet) โ‰ฅ 0. ฮป = (trace ยฑ โˆš(traceยฒโˆ’4ยทdet)) / 2.