Question

Without expanding the determinant, prove that

$\left|\begin{array}{ccc}a& a^2& bc\\ b& b^2& ca\\ c& c^2& ab\end{array}\right|=\left|\begin{array}{ccc}1& a^2& a^3\\ 1& b^2& b^3\\ 1& c^2& c^3\end{array}\right|$

Answer 1

$LHS=\left|\begin{array}{ccc}a& a^2& bc\\ b& b^2& ca\\ c& c^2& ab\end{array}\right|$
Operating $R_1\to a{R}_1,R_2\to b{R}_2$ and $R_3\to c{R}_3$ we obtain
$=\frac{1}{abc}\left|\begin{array}{ccc}{a}^2& a^3& abc\\ b^2& b^3& abc\\ c^2& c^3& abc\end{array}\right|=\frac{1}{abc}\times abc\left|\begin{array}{ccc}{a}^2& a^3& 1\\ b^2& b^3& 1\\ c^2& c^3& 1\end{array}\right|$ [Taking out factor abc from $C_3$]
$=(-1{)}^2\left|\begin{array}{ccc}1& a^2& a^3\\ 1& b^2& b^3\\ 1& c^2& c^3\end{array}\right|$ (using $C_1\leftrightarrow C_3$ and $C_2\leftrightarrow C_3$)
$=\left|\begin{array}{ccc}1& a^2& a^3\\ 1& b^2& b^3\\ 1& c^2& c^3\end{array}\right|=RHS$

Note: When we multiply any row or column of a determinant by any constant k then it is necessary that determinant divides by k.