Question

Without expanding the determinant, prove that $\left|\begin{array}{ccc}ax&by&cz\\x^{2}&y^{2}&z^{2}\\1&1&1\end{array}\right|$ = $\left|\begin{array}{ccc}a&b&c\\x&y&z\\yz&zx&xy\end{array}\right|$

Answer 1

HELLO DEAR,



GIVEN:- $\bold{\left|\begin{array}{ccc}ax&by&cz\\x^{2}&y^{2}&z^{2}\\1&1&1\end{array}\right|}$

=> $\bold{\left|\begin{array}{ccc}ax&by&cz\\x^{2}&y^{2}&z^{2}\\1&1&1\end{array}\right|}$


taking x , y & z common from C1 , C2 , C3 respectively,
we get,

=> $\bold{xyz\left|\begin{array}{ccc}a&b&c\\x&y&z\\(1/x)&(1/y)&(1/z)\end{array}\right|}$

multiply xyz with R3

=> $\left|\begin{array}{ccc}a&b&c\\x&y&z\\yz&zx&xy\end{array}\right|$



hence, $\bold{\left|\begin{array}{ccc}ax&by&cz\\x^{2}&y^{2}&z^{2}\\1&1&1\end{array}\right|}$ = $\bold{\left|\begin{array}{ccc}a&b&c\\x&y&z\\yz&zx&xy\end{array}\right|}$



I HOPE IT'S HELP YOU DEAR,
THANKS