> [!Note] Lemma (Uniqueness of inverse) > If $A\in \text{Mat}_{nn}$ has an inverse, then it is unique. We write $A^{-1}$ for this inverse. **Proof**: Suppose $B,C\in \text{Mat}_{nn}$ are inverses for $A$. Then by [[Associativity of Multiplication of Real Matrices|associativity of matrix multiplication]] and [[Product with Real Identity Matrix]], $C=I_{n}C=(BA)C=B(AC)=BI_{n}=B.$ **Proof**: Since $A\in \text{GL}_{n}(\mathbb{R}),$ it follows from [[General Linear Group Forms a Group]] and [[Uniqueness of Group Inverses]].