A sub category of row echelon form with every leading term equal to 1, and every term above it equal to 0. $\huge\rowechelon{ 1&0&0&a\\ 0&1&0&b\\ 0&0&1&c\\ }$ Finding a [[Reduced Row Echelon Form|RREF]] similar to the augmented matrix of a [[System of Linear Equations#Solving Linear Equations|System of Linear Equations]] will allow you to solve for $x,y,z$ (or however many variables are in the system of equations). All [[Equivalent]] [[Augmented Matrix|Augmented Matrices]] have a single [[Reduced Row Echelon Form|RREF]]. ### Special Cases #### Under-determined A [[System of Linear Equations]] with less equations than unknown [[Variable|Variables]] is [[Under-determined]], and must have either [[#Infinite Solutions]], [[#No Solutions]], or *multiple* solutions. #### Infinite Solutions In the case that the [[Reduced Row Echelon Form]] of an [[Augmented Matrix]] ends up with rows that are all zeroes, for every all zero row you introduce a [[Free Parameter]], and your [[Augmented Matrix]] has an infinite number of solutions. >[!example]- Free Parameter [[Reduced Row Echelon Form|RREF]] >$ >\begin{align} >\augmented{cccc|c} { >1&0&1&1&1\\ >0&1&2&1&2\\ >0&0&0&0&0\\ >0&0&0&0&0 >} &\sim >\begin{cases} >x_{1}+x_{3}+x_{4}&=1\\ >x_{2} +2x_{3}+x_{4} &= 2\\ >0 &= 0\\ >0 &= 0 >\end{cases}\\ >& \sim >\begin{cases} >x_{1}=1-x_{3}-x_{4}\\ >x_{2} = 2-2x_{3}-x_{4} >\end{cases}\\ >&\sim > >\boxed{ >\begin{cases} >x_{1}=1-t-s\\ >x_{2}=2-2t-s\\ >x_{3}=t\\ >x4=s >\end{cases} >} >\end{align} >$ #### No Solutions If the [[Reduced Row Echelon Form]] of an [[Augmented Matrix]] has a row where the leading term is at the very right, that means the [[System of Linear Equations]] is [[Inconsistent]] and has *no solutions*. >[!example]- [[Reduced Row Echelon Form|RREF]] With no Solutions >$\begin{align} >\augmented{ccc|c}{ >1&-2&3&-\frac{1}{2} \\ >0&0&0 & \frac{3}{2} \\ >0&0&0&1 >} >\implies >\boxed{0=1} >\end{align}$