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