Question

If the system of equations x + ay = 0, az + y = 0, ax + z = 0 has infinitely many solutions then a = ___________________.

Answer 1

The given system of homogeneous equations x + ay = 0, az + y = 0, ax + z = 0 has infinitely many solutions.

$\therefore \left|\begin{array}{ccc}1& a& 0\\ 0& 1& a\\ a& 0& 1\end{array}\right|=0$

$\Rightarrow 1\left(1-0\right)-a\left(0-a^2\right)+0\left(0-a\right)=0$

$\Rightarrow 1+a^3=0$

$\Rightarrow \left(1+a\right)\left(1-a+a^2\right)=0$

$\Rightarrow a+1=0\left(a^2-a+1>0\forall a\in \mathrm{R}\right)$

$\Rightarrow a=-1$

Thus, the value of a is −1.

If the system of equations x + ay = 0, az + y = 0, ax + z = 0 has infinitely many solutions then a = __−1__.