Matrix Rank Calculator

Find the rank of a matrix and see every row-reduction step. Exact fractions, no rounding. Dimensions up to 6×6.

rank = number of pivots after row reduction 1 1 pivots rank = 0

How to Find the Rank of a Matrix

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

1

Set the dimensions

Pick the rows and columns for A, from 1×1 up to 6×6. Rank is defined for any rectangular matrix, not just square ones.

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 rank and steps

The rank and row-echelon form appear first, followed by every row operation used to reach it — swaps, scalings, and eliminations.

Frequently Asked Questions

What is the rank of a matrix?
The rank of a matrix is the number of linearly independent rows (or equivalently, columns) it has. It equals the number of nonzero rows left after reducing the matrix to row-echelon form. Rank tells you the dimension of the space spanned by the matrix's rows or columns.
How do you find the rank of a matrix?
Row-reduce the matrix using elementary row operations (swap rows, scale a row, subtract a multiple of one row from another) until it is in row-echelon form. The rank is the number of nonzero rows, which equals the number of pivot columns found. This calculator performs and shows every row operation used.
What does a rank-deficient matrix mean?
A matrix is rank-deficient when its rank is less than the smaller of its row count and column count — some rows or columns are linear combinations of others. For a square matrix, being rank-deficient means it is singular: its determinant is zero and it has no inverse.
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... Row reduction involves many divisions and subtractions, and rounding errors can make a matrix that is truly rank-deficient appear full rank, so exact fractions keep the rank determination reliable.