> [!Note] Lemma (product of invertible matrices is invertible) > > If $A,B\in \text{Mat}_{mn}$ are invertible then $AB$ is invertible with $(AB)^{-1}=B^{-1}A^{-1}$. **Proof**: $(AB)(B^{-1}A^{-1})=A(B^{-1}B)A=AI_{n}A^{-1}=A A^{-1}=I_{n}$. Similarly $(B^{-1}A^{-1})(AB)=I_{n}$. **Proof**: Since $A\in \text{GL}_{n}(\mathbb{R}),$ it follows from [[General Linear Group Forms a Group]] and [[Inverse of the Product of Group Elements]].