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