Exemplify when an augmented matrix with pivot every row can still be consistent.

Question

While I read I come across:

In the 'warning' section, it claims an augmented matrix with pivot every row 'may be consistent'. However, I cannot think of any example or cases when this stands, given we don't consider 0 rows having any pivot.

Can anyone give me an example supporting this claim? Thanks in advance!

Answer 1

An example of an augmented matrix with pivots in each row with no solution:

$$\begin{pmatrix} 0 & | & 1 \end{pmatrix}$$

An example of an augmented matrix with pivots in each row with a solution:

$$\begin{pmatrix} 1 & | & 0 \end{pmatrix}$$