What you have to show is that if is a matrix which is not invertible and such that a principal submatrix of size is invertible then the rank of is . After permuting if necessary the rows and the columns, you can assume that it's the submatrix after deleting the -th row and -th column. Let the submatrix, and consider the blockwise matrix . Multipliying by , we can see that the rank of is at least . It can't be more since is not invertible.