Exercise 3.3.32

factors into

and the rank is $r=2.$ The special solution to $\mathit{Ax}=0$ and $\mathit{Ux}=0$ is $s=(-7,2,1).$ Since $b=(1,3,6,5)$ is also the last column of $A$, a particular solution to $Ax=b$ is $(0,0,1)$ and the complete solution is $x=(0,0,1)+cs$. (Or use the particular solution $x_p=(7,-2,0)$ with free variable $x_3=0.)$

For $b=(1,0,0,0)$ elimination leads to $Ux=(1,-1,0,1)$ and the fourth equation is $0=1$. No solution for this $b$.