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^{-1}A = I
Example: | |
and A^{-1}A = . |
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: |
Use Gauss-Jordan elimination to transform |
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 . |
See also
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