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

Answer: 6

For non-trivial solutions,
$\left|\begin{array}{ccc}a& 1& b\\ b& 1& a\\ a& b& ab\end{array}\right|=0$
$\Rightarrow \left(a-b\right)\left(a-b^2\right)=0$
$\Rightarrow$ 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 $6$