> [!NOTE] Lemma > Let $A$ be [[Real Square Matrices|real square matrix]] with [[Inverse of Real Square Matrix|inverse]] $A^{-1}.$ Then its [[Left Multiplication Linear Map of Real Matrix|left multiplication linear map]] $L_{A}$ is [[Bijection|bijective]] and its [[Function Inverse|inverse]] is given by $(L_{A})^{-1}=L_{A^{-1}}.$ **Proof**: Let $\underline{v}\in \mathbb{R}^{n}.$ By [[Left Multiplication Linear Map of Real Matrix Product]], $L_{A}(L_{A^{-1}}(\underline{v}))=L_{A A^{-1}}(\underline{v})=(AA^{-1})\underline{v} =I_{n} \underline{v}=\underline{v}=I_{n}\underline{v}=(A^{-1}A)\underline{v}=L_{A^{-1}A}(\underline{v} ) = L_{A^{-1}}(L_{A}(\underline{v})).$