# Submatrix rank question

If an $m\times n$ matrix with $m>n$ has full rank, does that imply that that uppermost $n\times n$ submatrix is invertible (also has full rank)?

If not, is it true at least that some combination of $n$ rows can be taken from the matrix to form an invertible matrix? How would one go about doing so?

• BTW, the study of which subsets of vectors are independent are precisely what led to the development to matroid theory. – p.s. Jun 29 '14 at 3:06

One thing to note is that an $m \times n$ matrix with $n>m$ cannot have rank $n$, since otherwise we would have $n$ linearly independent vectors in $\mathbb{R}^m$. Having full rank (in this case rank $m$) implies that there is some $m \times m$ invertible submatrix, but this is not necessarily the uppermost submatrix.