Question

If the system of equations $x+ay=0,az+y=0,ax+z=0$ has
infinite number of solutions then the value of $a$ is

• A. $-1$
• B. $1$
• C. $0$
• D. No real value

Answer 1

The correct option is A $-1$

$\begin{array}{}x+ay=0& \text{(1)}\end{array}$
$\begin{array}{}az+y=0& \text{(2)}\end{array}$
$\begin{array}{}ax+z=0& \text{(3)}\end{array}$
The above equations is a system of homogenous equations so, it will have infinite many solution if the determinant of the coefficient matrix is zero
$\left|\begin{array}{ccc}1& a& 0\\ 0& 1& a\\ a& 0& 1\end{array}\right|=0$
$\Rightarrow 1(1-0)-a(0-a^2)=0\Rightarrow a^3=-1\Rightarrow a=-1$