What is the rank of a matrix?

The rank is the maximum number of linearly independent rows (or columns) in a matrix. It equals the number of non-zero rows in its row echelon form and tells you the dimension of the row space.

What is Gaussian elimination?

Gaussian elimination is a systematic algorithm that transforms a matrix into row echelon form using three elementary row operations: swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another.

Can the rank exceed the number of rows or columns?

No. The rank of an m×n matrix is always at most min(m, n). A matrix with more columns than rows can have rank at most equal to the number of rows.

What does a rank-deficient matrix mean?

A matrix is rank-deficient when its rank is less than the smaller of its row or column count. This implies the rows (or columns) are linearly dependent, and a square rank-deficient matrix is not invertible (singular).

Does this handle decimal or negative entries?

Yes. You can enter any real number including decimals (e.g. 3.14) and negative values (e.g. -7). Results are displayed rounded to 4 significant figures to avoid floating-point noise.

Is there a limit on matrix size?

This tool supports matrices from 1×1 up to 5×5. For larger matrices you would typically use dedicated software such as NumPy or MATLAB.

Matrix Rank Calculator

Name: Matrix Rank Calculator
Availability: InStock
Author: Nham Vu

Enter a matrix (up to 5x5), and instantly compute its rank with a full step-by-step Gaussian elimination walkthrough.

Matrix Input

Rows

Columns

Enter a matrix and click Calculate Rank to see results.

Summary

Enter a matrix (up to 5x5), and instantly compute its rank with a full step-by-step Gaussian elimination walkthrough.

How it works

Select the number of rows and columns (up to 5x5).
Enter the matrix entries in the grid — decimals and negatives are supported.
Click "Calculate Rank" to run Gaussian elimination.
The tool reduces the matrix to row echelon form, recording every pivot and row operation.
Count of non-zero rows in the echelon form equals the rank, shown at the top of the results.
Expand the step-by-step log to review every individual row operation.

Use cases

Verify the rank of a coefficient matrix when solving a system of linear equations.
Check whether a set of vectors is linearly independent.
Determine the dimension of the column space or null space of a matrix.
Confirm that a matrix is full rank before computing its inverse.
Homework and exam prep for linear algebra courses.
Quick sanity-check for numerical linear algebra code.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu