Let $A \in S_n^{++}(\mathbf{R})$ and let $g : t \in \mathbf{R} \mapsto \operatorname{det}(I_n + tA)$. Express, for all $t \in \mathbf{R}$, $g(t)$ using the eigenvalues of $A$. Deduce that $g$ is of class $C^\infty$ on $\mathbf{R}$.
Let $A \in S_n^{++}(\mathbf{R})$ and let $g : t \in \mathbf{R} \mapsto \operatorname{det}(I_n + tA)$. Express, for all $t \in \mathbf{R}$, $g(t)$ using the eigenvalues of $A$. Deduce that $g$ is of class $C^\infty$ on $\mathbf{R}$.