Question

The number of solutions of the following equations $x_2-x_3=1,-x_1+2x_3=-2,x_1-2x_2=3$ is.

• A. Zero
• B. One
• C. Two
• D. Infinite

Answer 1

The correct option is A Zero

Let $A=\left[\begin{array}{ccc}0& 1& -1\\ -1& 0& 2\\ 1& -2& 0\end{array}\right]$
Now, $|A|=\left|\begin{array}{ccc}0& 1& -1\\ -1& 0& 2\\ 1& -2& 0\end{array}\right|$
$=0\left|\begin{array}{cc}0& 2\\ -2& 0\end{array}\right|-1\left|\begin{array}{cc}-1& 2\\ 1& 0\end{array}\right|-1\left|\begin{array}{cc}-1& 0\\ 1& -2\end{array}\right|$
$=0+2-2$
$=0$
$\Rightarrow |A|=0$
and $D_2=\left|\begin{array}{ccc}0& 1& -1\\ -1& -2& 2\\ 1& 3& 0\end{array}\right|$
$=0-1(0-2)-1(-3+2)$
$=2+1=3$
i.e., $D_2\ne 0$
$\therefore$ This system of equation is inconsistent. So, it has no solution. i.e, number of solutions is zero.