Matrix Inverse

A matrix A transforms x into Ax — it maps input to output. For a square matrix, sometimes there's a perfect undo: another matrix A⁻¹ that sends Ax back to x. With it, solving Ax = b is a one-liner: x = A⁻¹b. Cryptography uses inverses to decode; graphics uses them to reverse rotations; statistics uses them to fit models. But not every matrix has an inverse — and predicting which is one of the great dividing lines in linear algebra.