Heard that Hadamard's inequality:
$$\left|\det(A)\right| \leq {\prod_{i=1}^{n}\sqrt{\sum_{j=1}^{n} |a_{ij}|^2} } $$
can be proved by the use of Lagrange multiplier methods. I saw and understand the proof by the use of methods from linear algebra. However, I do not see how to apply such tool from mathematical analysis to detive the aforementioned proof.
If anyone would be able to provide proof, which uses Lagrange multiplier method, I would be very thankful!