Let $M \in S_n^+(\mathbf{R})$ be a non-zero matrix. Show the inequality $\frac{\operatorname{Tr}(M)}{n} \geq \operatorname{det}^{1/n}(M)$. Hint: You may show that $x \mapsto -\ln(x)$ is convex on $\mathbf{R}_+^\star$.
Let $M \in S_n^+(\mathbf{R})$ be a non-zero matrix. Show the inequality $\frac{\operatorname{Tr}(M)}{n} \geq \operatorname{det}^{1/n}(M)$.
Hint: You may show that $x \mapsto -\ln(x)$ is convex on $\mathbf{R}_+^\star$.