Question:
$3 x+y-2 z=0$
$x+y+z=0$
$x-2 y+z=0$
Solution:
The given system of homogeneous equations can be written in matrix form as follows:
$\left[\begin{array}{ccc}3 & 1 & -2 \\ 1 & 1 & 1 \\ 1 & -2 & 1\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$
or, $A X=O$
where, $A=\left[\begin{array}{ccc}3 & 1 & -2 \\ 1 & 1 & 1 \\ 1 & -2 & 1\end{array}\right], X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right]$ and $O=\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$
Now,
$|A|=\left|\begin{array}{ccc}3 & 1 & -2 \\ 1 & 1 & 1 \\ 1 & -2 & 1\end{array}\right|$
$=3(1+2)-1(1-1)-2(-2-1)$
$=9-0+6$
$=15 \neq 0$
So, the given system has only trivial solution, which is given below:
$x=y=z=0$