# Matrix Inverter

A matrix inverse calculator using Gauss-Jordan algorithm.

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

#### Timur

Created: 2011-03-28 12:56:34, Last updated: 2023-05-05 06:50:48

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This calculator computes the inverse matrix of any given matrix using the Gauss-Jordan algorithm. While it is possible to find the inverse of a small matrix (up to 4x4) manually (see the adjugate matrix section below the calculator), for larger matrices, a calculator that uses the Gauss-Jordan algorithm is necessary.

The Gauss-Jordan algorithm is an extension of the Gaussian elimination algorithm used to solve a system of linear equations. It is an iterative process where the given matrix is transformed into an identity matrix by performing elementary row operations. In the end, the identity matrix is transformed into the inverse matrix of the given matrix.

The algorithm works by combining the given matrix with an identity matrix to form an augmented matrix. The augmented matrix is then manipulated using elementary row operations to transform the given matrix into an identity matrix, while the identity matrix transforms into the inverse of the original matrix. If the original matrix is singular (non-invertible), the algorithm will terminate without finding an inverse matrix.

The inverse matrix can be used to solve systems of linear equations, and is used in many applications such as computer graphics, physics, and engineering.

#### Inverse of a matrix

Digits after the decimal point: 2
Inverse of a matrix

### Manually calculating the inverse matrix using the adjugate 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 the adjugate matrix to compute the matrix inverse using this formula:
$A^{-1} = \frac{1}{\det A}\cdot C^*$

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

The 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 easy 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, Matrix Inverter