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Matrix Rank Calculator

Find rank using RREF, view pivot columns and free variables, estimate nullity, and export results to CSV.

Rank RREF Pivots Nullity

Rank, RREF, Pivot Columns & Null Space Summary

Build a matrix, compute RREF, and read rank from the number of pivots.

How rank is computed here:
  • The calculator performs row reduction to Reduced Row Echelon Form (RREF).
  • A pivot is a leading 1 in RREF (after scaling rows).
  • Rank equals the number of pivots.
  • Nullity equals (number of columns − rank).
  • Pivot columns correspond to a basis for the column space of the original matrix.
Common uses of matrix rank:
  • Testing whether columns are independent (feature redundancy in data).
  • Checking if Ax=b has a unique solution (square matrices with full rank).
  • Understanding degrees of freedom (nullity/free variables).
  • Diagnosing singular matrices before computing an inverse.
  • Analyzing transformations (dimension of image / output space).

What Is Matrix Rank?

The rank of a matrix is the number of linearly independent rows or columns it contains. Informally, rank tells you how much “unique information” is in the matrix. If one row is a multiple of another, it does not add new information and does not increase rank. Rank is one of the most important diagnostics in linear algebra because it connects directly to invertibility, solution existence, and degrees of freedom.

Row Rank Equals Column Rank

A key theorem in linear algebra states that the number of independent rows equals the number of independent columns for any matrix. That means “row rank” and “column rank” are the same value. This calculator reports rank based on pivots in RREF, which is the standard, practical method in computation and coursework.

How RREF Reveals Rank

Reduced Row Echelon Form is a simplified version of the matrix obtained through elementary row operations (swap, scale, and add multiples). In RREF, each pivot column has a leading 1, and all other entries in that pivot column become 0. The number of pivot columns equals the number of pivots, which equals the rank.

This is why RREF is so useful: it not only gives rank, it shows which columns are pivots, which variables are free, and whether a related linear system is consistent.

Nullity and Free Variables

Nullity measures the size of the solution space of Ax=0. If a matrix has n columns and rank r, then nullity is n−r. Free variables correspond to non-pivot columns. In system-solving terms, free variables represent degrees of freedom: how many choices you can make while still satisfying the equations.

Rank and Invertibility

For a square n×n matrix, full rank means rank(A)=n. In that case det(A)≠0 and the matrix is invertible. If rank is less than n, the matrix is singular and cannot be inverted. Rank therefore provides a robust way to detect singularity, especially when determinants are very small due to floating-point rounding.

Rank in Data and Machine Learning

Rank is not only a classroom concept. In data science, rank indicates whether features are redundant. If columns are dependent, the design matrix is rank deficient, which can cause instability in regression or require regularization. Rank is also central to dimensionality reduction methods and to understanding the intrinsic dimension of datasets.

How to Use This Matrix Rank Calculator

  • Select the number of rows and columns (up to 10×10).
  • Build the matrix and enter values (or use Quick Fill for testing).
  • Click Calculate Rank to compute RREF, rank, pivot columns, and nullity.
  • Optionally export the input matrix or the computed RREF as CSV.

Accuracy Notes

Rank computation in floating-point arithmetic requires a tolerance because tiny rounding artifacts can appear. This tool treats values below a small threshold as zero. If your matrix is nearly singular, small changes can change pivots and therefore change the rank decision. In such cases, try using fewer decimals or scaling your inputs to similar magnitudes.

FAQ

Matrix Rank Calculator – Frequently Asked Questions

Answers about rank, RREF, pivots, nullity, and interpreting free variables.

Rank is the number of linearly independent rows or columns in a matrix. It equals the number of pivot positions in the matrix’s reduced row echelon form (RREF).

A common method is row reduction to RREF and counting pivots. This tool performs stable row reduction with pivoting and reports the pivot columns and rank.

Yes. For any matrix, the number of independent rows equals the number of independent columns, so row rank equals column rank.

Rank helps determine whether a system has a unique solution, infinitely many solutions, or no solution, depending on the rank of the coefficient matrix and augmented matrix.

Nullity is the dimension of the solution space of Ax=0. It equals the number of columns minus the rank (nullity = n − rank).

Pivot columns are the columns in the original matrix corresponding to leading 1s in RREF. They form a basis for the column space.

Floating-point arithmetic can produce small rounding artifacts. This tool zeroes values below a small tolerance, but nearly singular matrices can still show sensitivity.

This calculator supports up to 10×10 for rank and RREF in the browser. Very large matrices are better handled in dedicated linear algebra software.

Yes. You can export your input matrix and the computed RREF matrix to CSV for further analysis.

Rank and RREF are computed numerically with a tolerance. Nearly singular matrices can be sensitive to rounding, so interpret borderline cases carefully.

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