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## Matrix Inversion

Matrix inversion plays a major role in many multivariate techniques. While the inverse of a matrix is defined only for quadratic matrices, the concept of matrix inversion can also be extended to rectangular matrices by introducing the pseudo-inverse of a matrix.

 Inverse Matrix Given a square matrix A, the inverse matrix X is defined by the following equation: A X = IThe inverse of A is denoted by A-1 and is unique. Note that not all square matrices can be inverted. If an inverse for A exists, Ais called a regular or nonsingular matrix, otherwise A is a singular matrix. For regular matrices, the following equation is valid: A A-1 = A-1 A = I.

Please note that (A B)-1 is not equal to A-1 B-1, but rather to B-1 A-1. The inverse of a matrix may be calculated using several algorithms, one of them being the Gauss Jordan algorithm.

Example: Inverse Matrix since Last Update: 2005-Jän-25