Matrix Inverse Calculator

Find the inverse of a square matrix using Gauss-Jordan elimination, with every row operation shown. Exact fractions, no rounding. Dimensions up to 6×6.

A · A⁻¹ = I · = 1 0 0 1 A A⁻¹ I det = 0 invertible

How to Find a Matrix Inverse

Enter the entries of a square matrix — whole numbers, decimals, and fractions like 1/2 are all accepted.

1

Set the size

Pick a size from 1×1 to 6×6. Only square matrices have an inverse, so rows and columns always match.

2

Fill in the entries

Type a value into each cell. The result updates as you type, and every value is kept as an exact fraction rather than a rounded decimal.

3

Read the result and steps

The determinant and inverse appear first, followed by every Gauss-Jordan row operation applied to the augmented matrix. If the determinant is zero, the matrix is singular and has no inverse.

Frequently Asked Questions

How do you find the inverse of a matrix?
Augment the matrix with the identity matrix of the same size, then use row operations (swap rows, scale a row, subtract a multiple of one row from another) to reduce the left half to the identity matrix. Whatever the right half becomes is the inverse. This is called Gauss-Jordan elimination. This calculator performs and shows every one of those row operations.
Why do some matrices not have an inverse?
A matrix has an inverse only if it is square and its determinant is not zero. A matrix with a zero determinant is called singular: its rows are linearly dependent, so the Gauss-Jordan elimination cannot reduce the left half to the identity matrix, and no inverse exists.
How do you check that a matrix inverse is correct?
Multiply the original matrix by the computed inverse (in either order). If the inverse is correct, the product is the identity matrix — ones on the diagonal and zeros everywhere else. You can verify this using the Matrix Multiplication Calculator with the original matrix and this calculator's result.
Why does this calculator use fractions instead of decimals?
Matrix entries are stored and combined as exact fractions using integer arithmetic, so results like 1/3 stay exact instead of rounding to 0.333... Gauss-Jordan elimination involves many divisions and subtractions, and rounding errors compound quickly, so exact fractions keep every step and the final inverse mathematically precise.