We say two matrices are [[Equivalent]] if their system of linear equations have the same solution(s) > [!example] >$\huge\cases{ >x+y-3z=-2 \\ >-x+y-z=0\\ >2x+y+2z=9 >} >\sim\rowechelon{ >1&1&-3&-2\\ >-1&1&-1&0\\ >2&1&2&9 >}$ >[!seealso] See __[[Reduced Row Echelon Form#Special Cases|RREF]]__ for what different [[Augmented Matrix|Augmented Matrices]] tell us about the underlying [[System of Linear Equations]] ## Row Operations Row operations are operations you can do to an Augmented Matrix that creates a new [[Equivalent]] [[Matrix]]. - [[Augmented Matrix#Add Row|Take a row and add a multiple of it to another row]] - Multiply any *row* by some *nonzero* constant $C$ - Switch two rows >[!example]- Using [[Gauss-Jordan Elimination]] to turn an [[Augmented Matrix]] to [[Reduced Row Echelon Form|RREF]] >$\begin{align*} >\rowechelon{1&1&-3&-2\\-1&1&-1&0\\2&1&2&9} >&\sim\rowechelon{1&1&-3&-2\\0&2&-4&-2\\2&1&2&9} \\ &\sim\rowechelon{1&1&-3&-2\\0&2&-4&-2\\0&-1&8&13}\\ >&\sim\rowechelon{1&1&-3&-2\\0&1&-2&-1\\0&-1&8&13} \\ >&\sim\rowechelon{1&1&-3&-2\\0&1&-2&-1\\0&-1&8&13} \\ >&\vdots\\ >&\sim\rowechelon{1&0&0&1\\0&1&0&3\\0&0&1&2} >\end{align*}$ > >We stop when we get the left hand side to look like an [[Identity Matrix]], as this now corresponds to the equations $x=1$, $y=3$, $z=2$.