Prove that $\forall x , y \in \mathbb { R } ^ { n } , \left\langle A ^ { - 1 } x ; y \right\rangle = \left\langle x ; A ^ { - 1 } y \right\rangle$. Deduce that the matrix $A ^ { - 1 }$ is symmetric.
Prove that $\forall x , y \in \mathbb { R } ^ { n } , \left\langle A ^ { - 1 } x ; y \right\rangle = \left\langle x ; A ^ { - 1 } y \right\rangle$. Deduce that the matrix $A ^ { - 1 }$ is symmetric.