Editing Rank (linear algebra) (section)

=== Determinantal rank – size of largest non-vanishing minor ===
The rank of {{mvar|A}} is the largest order of any non-zero [[Minor (linear algebra)|minor]] in {{mvar|A}}.  (The order of a minor is the side-length of the square sub-matrix of which it is the determinant.) Like the decomposition rank characterization, this does not give an efficient way of computing the rank, but it is useful theoretically: a single non-zero minor witnesses a lower bound (namely its order) for the rank of the matrix, which can be useful (for example) to prove that certain operations do not lower the rank of a matrix.

A non-vanishing {{mvar|p}}-minor ({{math|''p'' × ''p''}} submatrix with non-zero determinant) shows that the rows and columns of that submatrix are linearly independent, and thus those rows and columns of the full matrix are linearly independent (in the full matrix), so the row and column rank are at least as large as the determinantal rank; however, the converse is less straightforward.  The equivalence of determinantal rank and column rank is a strengthening of the statement that if the span of {{mvar|n}} vectors has dimension {{mvar|p}}, then {{mvar|p}} of those vectors span the space (equivalently, that one can choose a spanning set that is a ''subset'' of the vectors): the equivalence implies that a subset of the rows and a subset of the columns simultaneously define an invertible submatrix (equivalently, if the span of {{mvar|n}} vectors has dimension {{mvar|p}}, then {{mvar|p}} of these vectors span the space ''and'' there is a set of {{mvar|p}} coordinates on which they are linearly independent).