Question

$\left|\begin{array}{ccc}2xy& x^2& y^2\\ x^2& y^2& 2xy\\ y^2& 2xy& x^2\end{array}\right|=$

• A. ${\left(x^3+y^3\right)}^2$
• B. ${\left(x^2+y^2\right)}^3$
• C. $-{\left(x^2+y^2\right)}^3$
• D. $-{\left(x^3+y^3\right)}^2$

Answer 1

Answer: (D)

Applying $C_1\to C_1+C_2+C_3$ , we get
$\Delta ={\left(x+y\right)}^2\left|\begin{array}{ccc}1& x^2& y^2\\ 1& y^2& 2xy\\ 1& 2xy& x^2\end{array}\right|$
Applying $R_2\to R_2-R_1,R_3\to R_3-R_1$, we get
$\Delta ={\left(x+y\right)}^2\left|\begin{array}{ccc}1& x^2& y^2\\ 0& y^2-x^2& 2xy-y^2\\ 0& 2xy-x^2& x^2-y^2\end{array}\right|$
$={\left(x+y\right)}^2\left[-{\left(x^2-y^2\right)}^2-\left(2xy-x^2\right)\left(2xy-y^2\right)\right]$
$=-{\left(x+y\right)}^2{\left[x^2-xy+y^2\right]}^2=-{\left(x^3+y^3\right)}^2$