----- Let $R$ be an [[integral domain]] and $\mathcal{M}_{m,n}(R)$ denote the [[vector space of m-by-n matrices|R-module of m-by-n matrices]]. Let $P,Q \in \mathcal{M}_{m,n}(R)$. > [!proposition] Proposition. ([[elementary operations produce equivalent matrices]]) > If $Q$ may be obtained from $P$ via a sequence of [[elementary operation|elementary operations]], then $P,Q$ are [[equivalent matrices]]. ^proposition > [!proof]+ Proof. ([[elementary operations produce equivalent matrices]]) > From [[change of basis formula]] and the resulting [[equivalent matrices|characterization of equivalent matrices]], it is enough to know that every elementary operation manifests as left or right multiplication by an invertible matrix — this is precisely the content of [[elementary operation]]. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```