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  4. The number of non-trivial solutions of the system x-y+z=0, x+2 y-z=0, 2 x+y+3 z=0 is

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Q. The number of non-trivial solutions of the system $x-y+z=0, x+2 y-z=0,\,2 x+y+3 z=0$ is

EAMCETEAMCET 2007

Solution:

Write given system of equations in matrix form $A X=B .$
$\begin{bmatrix}1 & -1 & 1 \\ 1 & 2 & -1 \\2 & 1 & 3\end{bmatrix}\begin{bmatrix}x \\y \\z\end{bmatrix}=\begin{bmatrix}0 \\0 \\0\end{bmatrix}$
Now,
$|A|=\begin{vmatrix}1 & -1 & 1 \\1 & 2 & -1 \\2 & 1 & 3\end{vmatrix}$
$=1(6+1)+1(3+2)+1(1-4)$
$=7+5-3 \\=9 \neq 0$
Since, $|A| \neq 0 .$ So, the given system of equations has only trivial solution.
So, there is no non-trivial solution.