Matrix Rank Calculator

How it works

1. 1Enter the matrix dimensions (rows and columns, each 1–4) and fill in the cells of matrix A.
2. 2The calculator row-reduces A to row echelon form using Gaussian elimination with partial pivoting.
3. 3The rank is returned as the count of non-zero rows — the number of linearly independent rows (and columns).

Use cases

• Checking whether a system of linear equations has a unique solution, infinitely many, or none.
• Determining the dimension of the column space or null space of a transformation matrix.
• Verifying that a set of vectors spans a subspace by testing whether their matrix has full rank.

Frequently asked questions

What does the rank of a matrix mean?

The rank is the number of linearly independent rows (equivalently columns). It equals the dimension of the column space and row space — the count of rows that cannot be written as combinations of the others.

What is the maximum possible rank for an m×n matrix?

The rank can never exceed min(m, n) — the smaller of the row count and column count. A matrix that reaches this maximum is called full rank.

How is rank computed by row reduction?

Gaussian elimination with partial pivoting reduces A to row echelon form, choosing the largest available pivot each step for stability. The rank equals the number of rows with a non-zero leading entry afterwards.