Chapter 2.4, Problem 57E

Guided Proof Prove that if $A$ is row-equivalent to $B$, and $B$ is row-equivalent to $C$, $A$ is row-equivalent to $C$.

Getting Started: to prove that If $A$ is row-equivalent to $C$, you have to find elementary matrices $E_1$, $E_2$…. $E_k$ such that $A=E_k\dots E_2{E}_{1}C$.

(i) Begin by observing that $A$ is row-equivalent to $B$ and $B$ is row-equivalent to $C$.

(ii) This means that there exist elementary matrices $F_1{F}_2\dots F_n$ and $G_1{G}_2\dots G_m$ such that $A=F_n\dots F_2{F}_{1}B$ and $B=G_m\dots G_2{G}_{1}C$.

(iii) Combine the matrix equations from step (ii).