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 :

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

$=\frac{1}{abc}\left|\begin{array}{ccc}{a}^2& a^3& abc\\ b^2& b^3& abc\\ c^2& c^3& abc\end{array}\right|$ $[R_1\to a{R}_1,R_2\to b{R}_2,R_3\to c{R}_3]$

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

$=\left|\begin{array}{ccc}1& a^2& a^3\\ 1& b^2& b^3\\ 1& c^2& c^3\end{array}\right|=RHS$ $[C_1\leftrightarrow C_3,C_2\leftrightarrow C_3]$