Question

The system of equations $-2x+y+z=a,x-2y+z=b,x+y-2z=c$ has

• A. no solution if $a+b+c\ne 0$
• B. unique solution if $a+b+c=0$
• C. infinite number of solutions if $a+b+c=0$
• D. none of these

Answer 1

Answer: (A), (C)

The correct options are
A no solution if $a+b+c\ne 0$
C infinite number of solutions if $a+b+c=0$

$\Delta =\left(\begin{array}{ccc}-2& 1& 1\\ 1& -2& 1\\ 1& 1& -2\end{array}\right)=(-6+3+3)=0$
${\Delta }_1=\left(\begin{array}{ccc}a& 1& 1\\ b& -2& 1\\ c& 1& -2\end{array}\right)=3(a+b+c)$
Similarly, ${\Delta }_2={\Delta }_3=3(a+b+c)$
So there will be no solution if $(a+b+c)\ne 0$ and infinite solutions if $(a+b+c)=0$