Inverse Calculator — Step-by-Step Solutions

Matrix Entries

a11

a12

a21

a22

Result

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Your calculations will appear here

The Inverse Calculator finds the inverse of matrices. Enter a square matrix and the calculator will compute its inverse if it exists, showing the result in a clear matrix format.

The calculator supports 2x2 and 3x3 matrices, checks whether the matrix is invertible (non-singular), and uses the adjugate method for exact results.

How to Use the Inverse Calculator

Select the matrix size (2x2 or 3x3).
Enter the matrix elements in the grid.
The calculator checks if the matrix is invertible and displays the inverse.

Formula and Theory

The inverse of a matrix A is a matrix A^(-1) such that A * A^(-1) = I (the identity matrix). A matrix has an inverse if and only if its determinant is non-zero.

For a 2x2 matrix [[a,b],[c,d]], the inverse is (1/det) * [[d,-b],[-c,a]] where det = ad - bc. For 3x3 matrices, the adjugate method or row reduction is used.

Worked Examples

Example 1: 2x2 inverse

Problem: Find the inverse of [[2, 1], [5, 3]]

Solution: det = 2*3 - 1*5 = 1. Inverse = [[3, -1], [-5, 2]]

Answer: [[3, -1], [-5, 2]]

Frequently Asked Questions

When does a matrix not have an inverse?

A matrix is not invertible (singular) when its determinant is zero. This means the rows (or columns) are linearly dependent.

What are matrix inverses used for?

Matrix inverses are used to solve systems of linear equations (Ax = b gives x = A^(-1)b), in computer graphics transformations, cryptography, and many engineering applications.