Equivalence of two-norm between a matrix and its absolute form

Question

Let $A$ a $n\times n$ real matrix, and denote by $|A|$ the matrix formed with the absolute values of the entries of A. I have proved that $||\; |A|\;||_2\le \sqrt{n}||A||_2$, using the identity $||A||_2^2\le ||A||_1||A||_{\infty}$, but I am not sure how I can prove that $$||\; |A|\;||_2\ge (1/\sqrt{n})||A||_2.$$ Please, any hint will be appreciated.