Rank is the number of linearly independent rows or columns in a matrix. It equals the number of non-zero rows in RREF. Max rank = min(rows, cols). A full-rank matrix has maximum possible rank.

How to find the rank of a matrix?

Reduce to RREF via Gaussian elimination. Count the number of non-zero rows. That is the rank. 2x2 identity has rank 2. Zero matrix has rank 0.

Can rank be greater than matrix size?

No. Rank is always <= min(rows, columns). A 2x3 matrix has max rank 2. A 3x2 matrix also has max rank 2. Full rank = min(m,n).

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Matrix Rank Calculator

Compute the rank via RREF for 2x2 and 3x3 matrices

Enter 2x2 Matrix

Rank Formula

Rank = number of non-zero rows in RREF

rank(A) <= min(rows, cols)

Full rank: rank = min(m,n)

Nullity = cols - rank

The rank of a matrix measures its non-degeneracy. It equals the dimension of the column space (or row space). Computing rank via RREF is the standard method. The rank-nullity theorem states that rank + nullity = number of columns.

⚠ A matrix has full rank if all rows and columns are linearly independent. Rank deficiency indicates linear dependence. For square matrices, full rank = non-singular = det != 0.

What is Matrix Rank?

The rank of a matrix is the maximum number of linearly independent row vectors (or column vectors). It is a fundamental measure of the matrix information content. A matrix with full rank has all rows/columns independent. A rank-deficient matrix has dependencies, meaning some information is redundant.

Row Rank

The number of linearly independent rows. Equals non-zero rows in RREF. Always equals column rank.

Full Rank

rank = min(m,n). All rows/cols independent. Square matrix is invertible. System Ax=b has unique solution.

Rank Deficient

rank < min(m,n). Some rows are linear combinations. Matrix is singular. Null space is non-trivial.

Rank-Nullity

rank(A) + nullity(A) = n (columns). Nullity is the dimension of the null space.

Teaching Example: 2x2 matrix A = [[1,2],[3,4]]
RREF: R2 = R2 - 3R1: [[1,2],[0,-2]]. R2 = R2/-2: [[1,2],[0,1]]. R1 = R1 - 2R2: [[1,0],[0,1]].
Non-zero rows = 2. Rank = 2 (full rank, invertible matrix).

Applications

Linear Systems Data Analysis Machine Learning Computer Vision Control Theory

Frequently Asked Questions

What is matrix rank?▼

Rank = number of linearly independent rows/columns. Equals non-zero rows in RREF. Max rank = min(m,n).

How to find the rank?▼

Convert to RREF via Gaussian elimination. Count non-zero rows. Identity matrix of size n has rank n.

What does rank tell us?▼

Full rank = invertible, unique solutions. Rank deficiency = singular. Nullity = cols - rank.

Can rank exceed matrix size?▼

No. Rank <= min(rows, cols). A 3x2 matrix has max rank 2. A 2x3 matrix also has max rank 2.

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