Question

If $A=\left[\begin{array}{ccc}1& a& bc\\ 1& b& ac\\ 1& c& ab\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right],$ where $a,b,c\ne 0,$ then the value of $det\left(A\right)-det\left(B\right)$ is

Answer 1

In $det\left(A\right)$ multiply by $a,b,c$ in $R_1,R_2,R_3$ respectively.
$\Rightarrow det\left(A\right)=\frac{1}{abc}\left|\begin{array}{ccc}a& a^2& abc\\ b& b^2& abc\\ c& c^2& abc\end{array}\right|$
Taking $abc$ common from $C_3$
$det\left(A\right)=\left|\begin{array}{ccc}a& a^2& 1\\ b& b^2& 1\\ c& c^2& 1\end{array}\right|$
Interchanging column $C_1$ and $C_2$ followed by $C_1$ and $C_3,$ we get
$\Rightarrow det\left(A\right)=\left|\begin{array}{ccc}1& a& a^2\\ 1& b& b^2\\ 1& c& c^2\end{array}\right|=det\left(B\right)$
$\Rightarrow det\left(A\right)-det\left(B\right)=0$