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. Nonsquare 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^{1}A
= I
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 GaussJordan 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 .

See
also
Determinant of a
matrix, cofactor
