# Ranks of a matrix and determinants of its minors

Given a $n\times n$ matrix M. Prove that M has rank less than $k$ if and only if all of the determinants of its $k\times k$ minors are $0$.

My progress: I have thought about this problem for a while, but I only got that if the $k*k$ minors are $0$, then the $k$ columns of that submatrix are linearly dependent. I don't see how to relate this fact to the linear dependence of the original $k$ columns of matrix M. Can anyone please help me with a detailed explanation?

• Many thanks for your hint! But I think it only explains the fact that the submatrix will have $k$ linearly dependent columns/rows when all the determinants of $k\times k$ minors are 0 (since column rank=row rank). But how to prove that these columns/rows, when extended to the columns of matrix M, are still linearly dependent? Do we need to expand the det(M) along some rows above all those $k$ linearly dependent rows to show that det(M)=0? – user177196 Dec 12 '14 at 21:48
• If there is a $k\times k$ minor of nonzero determinant, use row and column operations to turn it into a $k\times k$ identity matrix. Those $k$ columns will then be linearly independent. – Gyu Eun Lee Dec 12 '14 at 22:41
• The thing is, after obtaining that $k\times k$ identity matrix, how to prove that the new $k$ columns of matrix M is L.I? Since all we have now is the $k\times k$ identity matrix with $k$ LI columns, but this matrix is INSIDE the big matrix M. So we are not sure if these $k$ LI columns correspond to the $k$ LI columns of the big matrix M. If we can show such correspondence, the problem is solved, but I can't prove it. – user177196 Dec 13 '14 at 0:54
• Here's my attempt to prove that such correspondence exists: since these $k$ LI columns in the submatrix N will be part of $k$ big columns of M, if the $k$ big columns in M aren't LI, the $k$ columns in N are also not LI (contradiction). – user177196 Dec 13 '14 at 1:01
• Just think: can $(1,0,a)$ and $(0,1,b)$ ever be linearly dependent? – Gyu Eun Lee Dec 13 '14 at 1:06