Q113 / 170 · Manipal (MET) 2021 · Mathematics

The value of ' $a$ ' for which the system of equations $a x+y+z=0, x+a y+z=0, x+y+z=0$ possesses non-zero solutions are given by

  • A
    1,2
  • B
    $1,-1$
  • Correct answer
    C
    1
  • D
    None of these

Solution

Given, system of equations are $\begin{aligned} & a x+y+z=0, x+a y+z=0 \\ & \text { and } x+y+z=0 \text { possesses non-zero solution. }\end{aligned}$ $\therefore \quad\left|\begin{array}{lll}a & 1 & 1 \\ 1 & a & 1 \\ 1 & 1 & 1\end{array}\right|=0$ $\begin{array}{rr}\Rightarrow & a(a-1)-1(1-1)+1(1-a)=0 \\ \Rightarrow & a^2-a+0+1-a=0 \\ \Rightarrow & a^2-2 a+1=0 \\ \Rightarrow & (a-1)^2=0 \\ \Rightarrow & a=1\end{array}$

Hence, the correct answer is option: 'C'.

Relevant Questions