The real value of \( a \) for which the system of equations \( x+ay=0,y+az=0,z+ax=0 \) has infinitely many solutions, is :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → -1
The system is homogeneous, so it always has the solution x = y = z = 0. It has infinitely many solutions exactly when the coefficient determinant is zero:
\( \displaystyle \begin{vmatrix}1 & a & 0\\0 & 1 & a\\a & 0 & 1\end{vmatrix}=1(1-0)-a(0-a^2)+0 \)
\( \displaystyle =1+a^3 \)
\( \displaystyle 1+a^3=0 \)
\( \displaystyle a^3=-1 \)
\( \displaystyle a=-1 \quad (\text{real}) \)
Shortcut: Put a = −1: the equations become x = y, y = z, z = x, which hold for every x = y = z, so there are infinitely many solutions. For a = 1, 0 or 2, the determinant \( 1+a^3 \) is 2, 1 or 9, not zero.