 Solution

Let $[A\mid b]$ be the augmented matrix of the linear system. Consistency means that $b$ is in the column space of $A$, equivalently $\mathrm{rank}[A\mid b]=\mathrm{rank}A$. For a consistent system, choosing one solution $x_0$ gives every solution as $x_0+\ker A$. The rank-nullity theorem therefore gives the complete classification:

For the given equations, subtracting the first from the second and the second from the third givesIf $\lambda =0$, then $z=-1$ and the first equation gives $x=0$. If $\lambda =1$, the first reduced equation is $0=1$, so the system is inconsistent. ThusThese conclusions can also be checked from $\det A=1-\lambda$, but the row differences exhibit the inconsistency directly.