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Matrix Transformations & Determinants
It's useful to visualise a matrix transforming the plane, not just points in the plane.
Matrix controls

Edit the entries of [[a, b], [c, d]] and see what it does to space.

Presets

Input a point
det(A) = ad − bc
1.64
Singular / Invertible
Invertible
Orientation
preserved
Vector image
(2, 1) → (3.90, 2)
Plane
Transformed image is blue. The points Ai Aj and Av represent the inputted matrix A acting on the unit square where i = (1,0) and j = (0,1) as well as your inputted point
Original points
A i
A j
A v
Explanation

The first column of the matrix you input becomes the image of i = (1, 0) and the second column becomes the image of j = (0, 1).

The unit square has area 1. After the transformation it becomes a parallelogram, and its area is the determinant. That is why determinant measures area scale factor and whether orientation is preserved or reversed. This is all quite simple to visualise with the unit square, but gets tricky to understand when matrices act on matrices. I'll add that soon but I'm quite stumped on how to do it to be honest

det(A) = ad − bc

Areas are scaled by 1.64, with orientation preserved.

When det(A) = 0, the square collapses into a lower-dimensional shape. That is why singular matrices are not invertible: different points get squashed onto the same line or point, so the mapping cannot be uniquely reversed.