Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\varepsilon_0 > 0$ be such that for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$. We admit that the function $\Phi : t \mapsto ( A + t M ) ^ { - 1 }$ is of class $C ^ { 1 }$ on $] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [$. By noting that $\Phi ( t ) \times ( A + t M ) = I _ { n }$, show that $$\Phi ( t ) \underset { t \rightarrow 0 } { = } A ^ { - 1 } - A ^ { - 1 } M A ^ { - 1 } t + o ( t )$$
Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\varepsilon_0 > 0$ be such that for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$. We admit that the function $\Phi : t \mapsto ( A + t M ) ^ { - 1 }$ is of class $C ^ { 1 }$ on $] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [$. By noting that $\Phi ( t ) \times ( A + t M ) = I _ { n }$, show that
$$\Phi ( t ) \underset { t \rightarrow 0 } { = } A ^ { - 1 } - A ^ { - 1 } M A ^ { - 1 } t + o ( t )$$