inverse matrix

The inverse of a square matrix A, written A^(-1), is the matrix that undoes A. Apply A and then A^(-1) and you are right back where you started: A * A^(-1) = I and A^(-1) * A = I, where I is the identity. Think of it as the reverse gear for a transformation.

Not every matrix has one. The matrix must be square, and its determinant must be nonzero — such a matrix is called invertible or nonsingular. If A squashes space flat (determinant 0), information is lost and no undo exists.

Inverses give a clean way to think about solving A*x = b: multiply both sides by A^(-1) to get x = A^(-1) * b. This is mostly a conceptual tool; in practice people usually solve the system with elimination rather than computing the full inverse, which is slower and less stable.