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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

.

 

See also

Determinant of a matrix, cofactor

 


  this page updated 27-aug-12
Mathwords: Terms and Formulas from Algebra I to Calculus
written, illustrated, and webmastered by Bruce Simmons
NCTM Web Bytes December 2004 Web Bytes March 2005 Web Bytes