Why are all homogenous systems consistent? A linear system of form $A\vec{x}=\vec{0}$ is called homogeneous. Why are all homogenous systems consistent?

A system is defined as inconsistent if its row-reduced echelon form contains a row of form $\begin{bmatrix}
0 & 0 & 0 & ... & 0 & | & k 
\end{bmatrix}$ where $k \neq 0$ and | is a separator within augmented matrix. Since your system equals $\vec{0}$, it is impossible to have $k \neq 0$, rendering the system consistent.
