Exercise 3.4.14

It’s enough to check that $A$ satisfies the definition of reduced echelon form on the page 185. For the first condition, a nonzero row in $A$ is a nonzero row in $(A|b)$. So it will precede all the zero rows in $(A|b)$. But there may be some zero rows in $A$ who are nonzero rows in $(A|b)$. This kind of rows will be behind those nonzero row by the third condition for $(A|b)$. The second condition for $(A|b)$ implies the second condition for $A$. The third condition for $(A|b)$ implies the third condition for $A$.