Last updated: August 1, 2026
Matrix Rank Calculator
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Quick Answer
Enter a matrix in row format to reduce it to row-echelon form and count the pivots. The calculator reports the rank, computes nullity from the number of columns minus the rank, and includes the echelon form so you can inspect the pivot structure directly.
Enter a matrix in row form to compute its rank, its nullity, and its row-echelon form.
Creators
Dharmendra SinghReviewers
Creators
Dharmendra SinghReviewers
Formula
Rank is the number of pivots in row-echelon form, and nullity equals the column count minus the rank.
Where:
- A=input matrix
- rank(A)=pivot count
- nullity(A)=dimension of the null space
Worked Examples
Full-rank 2×2 matrix
Use [[1,2],[3,4]].
- 1Row-reduce the matrix.
- 2Keep both pivots.
- 3Compute nullity from the column count.
Dependent rows
Use [[1,2],[2,4]].
- 1Reduce the second row to zero.
- 2Count the remaining pivots.
Rectangular matrix
Rank works for 2×3 matrices too.
- 1Reduce the rectangular matrix.
- 2Use columns minus rank for nullity.
Identity matrix
Every row contributes a pivot.
- 1The matrix is already in echelon form.
- 2Rank matches the size.
Introduction
Find the rank, nullity, and row-echelon form of a matrix using elimination.
Overview
Find the rank, nullity, and row-echelon form of a matrix using elimination.
Rank
Independence tests
Redundancy detection
Null space preparation
Input format
Enter the matrix in row form, such as 1,2;3,4.
Separate rows with semicolons
Separate entries with commas or spaces
Brackets are optional
Keep row lengths consistent
Formula and method
Rank is the number of pivots in row-echelon form, and nullity equals the column count minus the rank.
- A matters:
input matrix.
- Rank(A) matters:
pivot count.
- Nullity(A) matters:
dimension of the null space.
Validation rules
The calculator uses deterministic validation before it computes the final result.
The matrix must be rectangular and nonempty
Every entry must be finite
Malformed rows are rejected
Rank is computed with pivoting-aware elimination
Interpreting the outputs
Use the main result first, then the secondary fields to verify the structure and meaning of the answer.
rank counts independent directions
nullity counts free variables in Ax = 0
rowEchelonForm shows the pivot structure
Rectangular matrices are supported
Common use cases
These are typical reasons to use this calculator in study or applied work.
Independence tests
Redundancy detection
Null space preparation
Invertibility checks for square matrices
FAQs
What is matrix rank?
Rank is the number of pivots after row reduction.
What is nullity?
Nullity is the number of columns minus the rank.
Can a rectangular matrix have full rank?
Yes. It can have full row rank or full column rank depending on its shape.
Why include row-echelon form?
It lets you inspect the pivots and compare the result with your own work.
Can I use decimals?
Yes. Any finite real entries are supported.
Does full rank always mean invertible?
Only for square matrices.