How this calculator works
Row reduction turns the matrix into row echelon form without changing its row space. The number of non-zero rows is the rank. Equivalently, the rank is the number of pivot columns, the largest number of independent rows or columns, or the dimension of the column space. The rank-nullity theorem says rank plus nullity equals the number of columns.
Formula
rank(A) + nullity(A) = number of columns of A. Rank is the size of the largest non-zero minor.
Worked example
The matrix with rows 1, 2 and 2, 4 has rank 1 because the second row is twice the first. Its nullity is 2 - 1 = 1. A 3 by 3 identity matrix has rank 3 and nullity 0.
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Frequently asked questions
Is rank the same for rows and columns?
Yes. The row rank and column rank of any matrix are equal, so either reduction direction gives the same number.
What does full rank mean?
A matrix has full rank when its rank equals the smaller of its row and column counts. For a square matrix, full rank is the same as being invertible.
How does tolerance affect the rank?
This tool treats pivots below a small relative tolerance as zero. Matrices with very small but non-zero pivots can therefore report a lower rank than exact arithmetic would.