A very solid start on understanding the four basic kinds of the Reduced Row Echelon Form of $\textbf{A}$, $\textbf{R}_\textbf{A}$. We gather the rank of $\textbf{A}$, whether solutions to $\mathbf{A}\vec{x}=\vec{b}$ are possible or not depending on $\textbf{b}$, and if so, whether there will be one or infinitely many. Yes, it's that exciting. Later we add $\textbf{R}_\textbf{A}^{\mbox{T}}$ to complete the Big Picture story (gloriousness).