Mathematics · Simulator 044
A 2×2 matrix moves every point of the plane. Edit its four entries, or pick a preset, and watch the grid stretch, rotate, shear or flip. The determinant tells you how areas change, and the eigenvectors are the directions that only get stretched.
The matrix M = [[a, b], [c, d]] sends a point (x, y) to (ax + by, cx + dy). Its columns are where the basis vectors land: î = (1, 0) goes to (a, c), and ĵ = (0, 1) goes to (b, d). Everything else follows, because grid lines stay straight, parallel and evenly spaced.
Press Play to animate smoothly from the identity matrix to M (linear interpolation of entries), which shows how each kind of transformation “moves”.
The animation interpolates matrix entries, which is not the same as rotating by a growing angle; a true rotation animation would pass through rotation matrices.
Find a matrix that doubles lengths along the line y = x and leaves the line y = −x unchanged. What is its determinant?
Eigenvalues 2 (along (1, 1)) and 1 (along (1, −1)) give M = [[1.5, 0.5], [0.5, 1.5]], with det = 2.25 − 0.25 = 2, the product of the eigenvalues. Enter it and check the eigenvector readout.