For an $n\times n$ positive definite matrix $A$, I wish to prove that 


  $$\det(A) \leq \bigg(\frac{Trace(A)}{n}\bigg)^n$$


To me this seems some form of AM-GM Inequality (Arithmatic Mean-Geometric Mean Inequality). Therefore If I can show the following, above inequality follows : 


  $$\det(A) \leq \prod_{i=1}^{i=n} A_{ii}$$


Any idea how to prove the above.
Thanks


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