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Prove $\det (A)\le \left(\max_{i,j}|A_{ij}|\right)^n n^{n/2}$

How do I prove this as an extension of the Hadamard Inequality? This expression is given as a corollary of the Hadamrd inequality in multiple places, but I'm not able to prove it.

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    $\begingroup$ Hint: the norm of $k^{th}$ column $= \sqrt{A_{1k}^2 + A_{2k}^2 + \cdots + A_{nk}^2} \le \sqrt{n} \max\limits_{1\le i \le n} |A_{ik}| \le \sqrt{n} \max\limits_{1 \le i, j \le n } |A_{ij}|$. $\endgroup$ – achille hui Oct 13 '16 at 12:25

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