What is the rank of a matrix?

Rank as the number of independent rows, how to compute it by elimination, and what full rank means for invertibility and linear systems.

The rank of a matrix is the number of linearly independent rows — equivalently, of linearly independent columns. Those two numbers are always equal, a fact important enough to have a name: the row rank equals column rank theorem.

Computing it

Row-reduce the matrix and count the non-zero rows. That count is the rank. Row operations never change the rank, which is exactly why this works.

The second row here was just twice the first, so it carries no new information and collapses to zero.

Bounds and full rank

For an m×n matrix, rank ≤ min(m, n). A matrix hitting that bound has full rank. For a square matrix, full rank is the same as being invertible, and the same as having a non-zero determinant — three different phrasings of one property.

Why it matters

• Rouché–Capelli theorem: the system Ax = b is consistent exactly when rank(A) = rank([A | b]). If that common rank equals the number of unknowns, the solution is unique; otherwise there are infinitely many, with n − rank free parameters.
• Rank-nullity: rank(A) + nullity(A) = n, the number of columns.
• Rank tells you the dimension of the image of the linear map, so it measures how much the transformation collapses space.
• rank(A·B) ≤ min(rank(A), rank(B)) — multiplying can never create independence.