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Inverse of a Matrix
Matrix Inverse
Multiplicative Inverse of a Matrix

For a square matrix A, the inverse is written A-1. When A is multiplied by A-1 the result is the identity matrix I. Non-square matrices do not have inverses.

Note: Not all square matrices have inverses. A square matrix which has an inverse is called invertible or nonsingular, and a square matrix without an inverse is called noninvertible or singular.

AA-1 = A-1A = I

 Example: For matrix , its inverse is since AA-1 = and A-1A = .

Here are three ways to find the inverse of a matrix:

 1. Shortcut for 2x2 matrices For , the inverse can be found using this formula: Example: 2. Augmented matrix method Use Gauss-Jordan elimination to transform [ A | I ] into [ I | A-1 ]. Example: The following steps result in . so we see that . 3. Adjoint method A-1 = (adjoint of A)   or   A-1 = (cofactor matrix of A)T Example: The following steps result in A-1 for . The cofactor matrix for A is , so the adjoint is . Since det A = 22, we get .