[Math] Non-trivial solution to a homogeneous system of linear equations.

Question

I have equations:
\begin{cases}
2x + y – z = 0\\
x – 2y – 3z =0\\
-3x – y + 2z =0
\end{cases}
After I put this in matrix row-reduced echelon form I get solutions of $x=0, y=0, z=0$. But my book says it has a non-trivial solution. Could someone explain how that could be?

Answer 1

Let $$A= \begin{pmatrix}2 & 1 & -1\\1 & -2 & -3\\ -3 & -1 & 2 \end{pmatrix},$$ and $v_i$, $i=1,2,3$ be the column vectors of $A$. Observe that $v_1-v_2+v_3=0$. This implies that $\{v_1,v_2,v_3\}$ is linearly independent, so $\ker A$ is nontrivial. In particular, $$\begin{pmatrix}1\\-1\\1\\\end{pmatrix}\in\ker A.$$