Exercise 16

A submatrix of a given matrix $A$ is one that we get by deleting some of the rows and columns of $A$. Thus, the first matrix here is a submatrix of the second.

Prove that for any square matrix, the rank of the matrix is $r$ if and only if $r$ is the largest integer such that there is an $r\!\times\!r$ submatrix with a nonzero determinant.