Matrices proofs. who needs them!?

Since m < n, then the rank of the matrix is m or less. The matrix (A^T)A is nxn, but its rank cannot be greater than the rank of A. Only matrices with full rank can have a non-zero determinant. Since (A^T)A is nxn and its rank is m or less and m < n, (A^T)A must be singular and its determinant is 0.