Consistent Systems

Augmented Matrices

This:

$$\begin{array}{c}\{\begin{array}{l}{x}_1-2x_2+x_3=0\\ 2x_2-8x_3=7\end{array}\end{array}$$

Becomes: $\left[\begin{array}{cccc}1& -2& 1& 0\\ 0& 2& -8& 7\end{array}\right]$ A linear system is consistent if it has at least one solution. Two matrices are row equivalent if a sequence of row operations transforms one matrix into the other.

A and B are row equivalent A and C are not row equivalent $\left[\begin{array}{cc}A& \mathbf{b}\end{array}\right]=\left[\begin{array}{ccc}1& 1& 1\\ 0& 1& 1\end{array}\right]$ $\left[\begin{array}{cc}A& \mathbf{b}\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\end{array}\right]$ So $\left[\begin{array}{cc}A& \mathbf{b}\end{array}\right]$ are consistent. And $\left[\begin{array}{cc}C& \mathbf{b}\end{array}\right]$ are not consistent.