# Inverse matrix calculator

This online calculator finds inverse matrix via adjugate matrix

### This page exists due to the efforts of the following people:

#### Timur

Created: 2011-03-28 12:56:34, Last updated: 2021-02-26 13:46:49

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The calculator below computes the inverse matrix via the Gauss-Jordan algorithm. You can find theory and formulas under the calculator.

#### Inverse of a matrix

Digits after the decimal point: 2
Inverse of a matrix

The inverse of a square matrix A, sometimes called a reciprocal matrix, is a matrix $A^{-1}$ such that
$AA^{-1} = A^{-1}A = I$

For manual calculation you can use adjugate matrix to compute matrix inverse like this:
$A^{-1} = \frac{1}{\det A}\cdot C^*$

Adjugate matrix is the transpose of the cofactor matrix of A.
${C}^{*}= \begin{pmatrix} {A}_{11} & {A}_{21} & \cdots & {A}_{n1} \\ {A}_{12} & {A}_{22} & \cdots & {A}_{n2} \\ \vdots & \vdots & \ddots & \vdots \\ {A}_{1n} & {A}_{2n} & \cdots & {A}_{nn} \\ \end{pmatrix}$

Cofactor of $a_{ij}$ of A is defined as
$A_{ij}=(-1)^{i+j}M_{ij}$
where $M_{ij}$ is a minor of $a_{ij}$.

You can use this method relatively easily for small matrices, 2x2, 3x3, or, maybe, 4x4. For bigger matrices, it is easier to use the Gauss-Jordan algorithm implemented by the calculator.

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PLANETCALC, Inverse matrix calculator