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

$\begin{array}{1 1}(A)\;0&(B)\;1\\(C)\;2&(D)\;3\end{array}$

Let us write the given system of equation in matrix form $Ax=B$
$\begin{bmatrix}1&-1&1\\1&2&-1\\2&1&3\end{bmatrix}\begin{bmatrix} x\\y\\z\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}$
Now $|A|=\begin{vmatrix} 1&-1&1\\1&2&-1\\2&1&3\end{vmatrix}$
$\Rightarrow 1(6+1)+1(3+2)+1(1-4)$
$\Rightarrow 7+5-3$
$\Rightarrow 9\neq 0$
Hence $|A|\neq 0$ so the given system of equations has only trivial solutions.So there is no non trivial solution.
Hence (A) is the correct answer.