Question

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

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (A)

Write given system of equations in matrix form $AX=B.$
$\left[\begin{array}{ccc}1& -1& 1\\ 1& 2& -1\\ 2& 1& 3\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
Now,
$|A|=\left|\begin{array}{ccc}1& -1& 1\\ 1& 2& -1\\ 2& 1& 3\end{array}\right|$
$=1(6+1)+1(3+2)+1(1-4)$
$$\begin{gathered} =7+5-3 \\ =9\ne 0 \end{gathered}$$
Since, $|A|\ne 0.$ So, the given system of equations has only trivial solution.
So, there is no non-trivial solution.