Question

If $f\left(x\right)=\left|\begin{array}{ccc}a& -1& 0\\ ax& a& -1\\ a{x}^2& ax& a\end{array}\right|$, using properties of determinets find the value of $f\left(2x\right)-f\left(x\right)$ ?

Answer 1

$fx)=\left|\begin{array}{ccc}a& -1& 0\\ ax& a& -1\\ a{x}^2& ax& a\end{array}\right|$

$f\left(2x\right)=\left|\begin{array}{ccc}a& -1& 0\\ 2ax& a& -1\\ 4a{x}^2& 2ax& a\end{array}\right|$

$f\left(2x\right)-f\left(x\right)=\left|\begin{array}{ccc}0& 0& 0\\ ax& 0& 0\\ 3a{x}^2& ax& 0\end{array}\right|=0$.