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 its column count, meaning some rows or columns are linear combinations of others. The matrix [[1, 2], [2, 4]] has rank 1 rather than 2, because the second row is exactly twice the first. For a square matrix, being rank-deficient is the same as being singular: the determinant is zero and no inverse exists. In a linear system it means the equations carry less independent information than there are unknowns.
Why does this calculator use fractions instead of decimals?
Matrix entries are stored and combined as exact fractions using integer arithmetic, so a value like 1/3 stays exact instead of rounding to 0.3333. Row reduction involves many divisions and subtractions, and rounding errors can leave a pivot at a tiny non-zero value instead of a clean 0. A matrix that is genuinely rank-deficient would then be counted as full rank, which is a wrong answer rather than a slightly imprecise one. Exact fractions make the rank determination reliable for every matrix you enter.

Related matrix & vector calculators

Matrix Multiplication

Multiply two matrices with the full step-by-step working.

Matrix & Vectors

→

Determinant Calculator

Compute the determinant of a square matrix, with cofactor-expansion steps.

Matrix & Vectors

→

Inverse Matrix Calculator

Find the inverse of a square matrix via Gauss-Jordan elimination.

Matrix & Vectors

→

Matrix Transpose

Transpose any matrix by swapping its rows and columns.

Matrix & Vectors

→

Solve Ax = b

Solve a linear system Ax = b with step-by-step elimination.

Matrix & Vectors

→

Eigenvalue Calculator

Eigenvalues of a 2×2 or 3×3 matrix.

Matrix & Vectors

→

Eigenvector Calculator

Find the eigenvectors of a 2×2 or 3×3 matrix for each real eigenvalue, unit-normalized, with the null-space steps shown.

Matrix & Vectors

→

Dot Product Calculator

Dot product, magnitudes and angle between two vectors.

Matrix & Vectors

→

Cross Product Calculator

Cross product of two 3-D vectors.

Matrix & Vectors

→

All Math Calculators

Algebra, matrices, statistics and geometry with step-by-step solutions.

Math

→