Matrix Rank Calculator
The rank of a matrix counts its linearly independent rows and tells you the true dimensionality of the information it carries. This calculator finds the rank of a 3x3 matrix by performing Gaussian elimination with partial pivoting to reach row echelon form, then counting the rows that remain nonzero. Enter the nine entries and the tool returns the rank, whether the matrix has full rank, and the resulting nullity. Rank is fundamental to solving linear systems, testing invertibility, and understanding the geometry of a linear transformation.
Rank by row reduction
1. Reduce matrix to row echelon form
2. Use partial pivoting to avoid tiny pivots
3. Count rows that are not all zero
rank = number of nonzero pivot rows
nullity = columns - rank
Gaussian elimination clears entries below each pivot in turn. A row whose entries all become negligible contributes no pivot. The number of surviving pivots is the rank by the rank-nullity theorem.
Notes on matrix rank
- Each row must contain exactly three comma-separated numbers.
- Rank ranges from 0 (the zero matrix) to 3 (full rank) for a 3x3 matrix.
- A full-rank 3x3 matrix has a nonzero determinant and is invertible.
- The nullity equals the number of columns minus the rank.
- Values smaller than a small tolerance are treated as zero to absorb rounding.
Matrix rank: frequently asked questions
What is the rank of a matrix?
The rank of a matrix is the number of linearly independent rows, equivalently the number of nonzero rows after reducing the matrix to row echelon form. It also equals the dimension of the column space.
How is matrix rank computed?
This calculator applies Gaussian elimination to bring the matrix to row echelon form, then counts the rows that are not entirely zero. Each such row contains a leading pivot, and the pivot count is the rank.
What is the maximum rank of a 3x3 matrix?
The maximum rank of a 3x3 matrix is 3, called full rank. A full-rank 3x3 matrix is invertible and has a nonzero determinant. A rank below 3 means the rows are linearly dependent and the determinant is zero.
What does a rank of zero mean?
A rank of zero occurs only for the zero matrix, in which every entry is zero. Any matrix with at least one nonzero entry has rank of at least one.
How does rank relate to invertibility?
A square matrix is invertible if and only if it has full rank, that is its rank equals its number of rows. A rank-deficient square matrix is singular and cannot be inverted.
Official sources
- NIST Digital Library of Mathematical Functions: Determinants and linear operators.
- NIST/SEMATECH e-Handbook of Statistical Methods: Linear algebra reference.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.