Since $A$ and $B$ are row-equivalent matrices, consideration of the three row operations (Definition RO) will show that the augmented matrices, $\augmented{A}{\zerovector}$ and $\augmented{B}{\zerovector}$, are also row-equivalent matrices. This says that the two homogeneous systems, $\homosystem{A}$ and $\homosystem{B}$ are equivalent systems. $\homosystem{A}$ has only the zero vector as a solution (Definition NM), thus $\homosystem{B}$ has only the zero vector as a solution. Finally, by Definition NM, we see that $B$ is nonsingular.
Form a similar theorem replacing “nonsingular” by “singular” in both the hypothesis and the conclusion. Prove this new theorem with an approach just like the one above, and/or employ the result about nonsingular matrices in a proof by contradiction.