Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by
$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$
Show that $f _ { A } ( t ) \underset { t \rightarrow 0 } { = } \operatorname { det } ( A ) + \operatorname { det } ( A ) \operatorname { Tr } \left( A ^ { - 1 } M \right) t + o ( t )$.
Hint: You may begin by treating the case where $A = I _ { n }$.