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