Matrix trace inequality proof

Question

I am stuck on this problem. I have no idea how to proceed, so any help would be welcome.

Let $A$ be symmetric positive definite matrix $n\times n$ prove that :
$$\operatorname{trace}(A) * \operatorname{trace}(A^{-1}) \ge n^2$$

Answer 1

Hint: Let $D$ be a $n\times n$ diagonal matrix with eigenvalues $0<\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$. Then we see that \begin{align} \operatorname{tr}(D)\operatorname{tr}(D^{-1}) =&\ \left(\lambda_1+\cdots +\lambda_n \right)\left(\frac{1}{\lambda_1}+\ldots+\frac{1}{\lambda_n} \right)\geq\ n^2 \end{align} where the last inequality follows from the AM-HM inequality.