rank

$\operatorname{rank}(A)$ — the shared dimension of the row space and column space.

For an $m\times n$ matrix $A$ , the rank is the number of pivot columns in $A$ 's RREF, equivalently $\dim(\operatorname{Row}(A)) = \dim(\operatorname{Col}(A))$ — the row space and column space live in different ambient spaces, yet always share the same dimension. $\operatorname{rank}(A)\le\min(m,n)$ always; a matrix achieving that maximum is full rank, and for a square matrix full rank is exactly the condition for invertibility.