Question

Consider the system of equations $ax+y+bz=0,bx+y+az=0$ and $ax+by+abz=0$ where $a,b\in \left\{0,1 , 2 , 3,4\right\}.$ The number of ordered pairs $\left(a , b\right)$ for which the system has non-trivial solutions is

Answer 1

For non-trivial solutions,
$\begin{vmatrix} a & 1 & b \\ b & 1 & a \\ a & b & ab \end{vmatrix}=0$
$\Rightarrow \left(a - b\right)\left(a - b^{2}\right)=0$
$\Rightarrow $ Either $a=b$ or $a=b^{2}$
When $a=b,$ then ordered pairs are $\left(0,0\right),\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(4,4\right)$
When $a=b^{2},$ then $\left(4 , 2\right)$
Hence, number of ordered pairs are $6$