Prove $|\det A| \leq \prod_{j=1}^n ||a_j||$

Question

Let's say A is a square n by n matrix. ||$x$||=$x^T x$ and x is a real n-column norm. How would you show this? I tried to use the QR factorization here in showing that ||$a_j$||=||$r_j$||, but I'm not sure how far that will get you.

Answer 1

Let $A=QR$ be the QR factorisation of $A$ with $R=[r_1,\ldots,r_n]$. Then Mr Hadamard says that $$ \left|\,\det A\,\right| = \left|\,\det QR\,\right| = \left|\,\det Q\,\right|\;\left|\,\det R\,\right| = \left|\,\det R\,\right| = \prod_{i=1}^n\left|\,r_{ii}\,\right| \leq \prod_{i=1}^n\|r_i\|=\prod_{i=1}^n\|a_i\|. $$