Question

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

Answer 1

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

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

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

Expanding along row $1,$ clearly shows that the value of the determinant is $1.$