Question

Consider the following system of equations :
$\alpha x+y+z=m$
$x+\alpha y+z=n$
$x+y+\alpha z=p$
Which of the following statements is (are) CORRECT?

• A. The given system of equations has no solution if $\alpha =-2$ and $m+n+p\ne 0$
• B. The given system of equations has no solution if $\alpha =1$ and $m\ne n$ or $n\ne p$ or $p\ne m$
• C. The given system of equations has infinite solutions if $\alpha =-2$ and $m+n+p=0$
• D. The given system of equations has unique solution if $\alpha =1$ or $-2$

Answer 1

Answer: (A), (B), (C)

The given system of equations has infinite solutions if $\alpha =-2$ and $m+n+p=0$

$\Delta =\left|\begin{array}{ccc}\alpha & 1& 1\\ 1& \alpha & 1\\ 1& 1& \alpha \end{array}\right|=(\alpha -1{)}^2(\alpha +2)$

${\Delta }_1=\left|\begin{array}{ccc}m& 1& 1\\ n& \alpha & 1\\ p& 1& \alpha \end{array}\right|=(\alpha -1)\left[m\right(\alpha +1)-(n+p\left)\right]$

${\Delta }_2=\left|\begin{array}{ccc}\alpha & m& 1\\ 1& n& 1\\ 1& p& \alpha \end{array}\right|=(\alpha -1)\left[n\right(\alpha +1)-(m+p\left)\right]$

${\Delta }_3=\left|\begin{array}{ccc}\alpha & 1& m\\ 1& \alpha & n\\ 1& 1& p\end{array}\right|=(\alpha -1)\left[p\right(\alpha +1)-(m+n\left)\right]$

If $\alpha =-2,$ then ${\Delta }_i=3(m+n+p)$
and if $m+n+p\ne 0,$ then system is inconsistent.

If $\alpha =1$ and $m\ne n\ne p$ (or any two are not equal), then system is inconsistent having no solution.

If $\alpha =-2$ and $m+n+p=0,$ then system is consistent having infinite solutions.

If $\alpha =1$ or $\alpha =-2,$ then unique solution is not possible.