Past exam of the options II course of the University of Oxford / 2013 / P4 / d

Use a QR factorization, whose existence is supplied. Since $|\det Q|=1$ and $R$ is triangular,Each diagonal entry is one component of its column, so $|r_{jj}|\le \Vert r_j\Vert =\Vert a_j\Vert$. Multiplication yields Hadamard determinant inequality:The argument includes singular matrices, since a zero diagonal factor makes the determinant zero.