Question

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

• 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 $C_{1} \rightarrow C_{1}+C_{2}+C_{3}$ , we get
$\Delta=\left(x+y\right)^{2} \left|\begin{matrix}1&x^{2}&y^{2}\\ 1&y^{2}&2 xy\\ 1&2 xy&x^{2}\end{matrix}\right|$
Applying $R_{2} \rightarrow R_{2}-R_{1}, R_{3} \to R_{3}-R_{1}$, we get
$\Delta=\left(x+y\right)^{2} \left|\begin{matrix}1&x^{2}&y^{2}\\ 0&y^{2} -x^{2}&2xy-y^{2}\\ 0&2xy-x^{2}&x^{2}-y^{2}\end{matrix}\right|$
$=\left(x+y\right)^{2}\left[-\left(x^{2}-y^{2}\right)^{2}-\left(2xy-x^{2}\right)\left(2xy-y^{2}\right)\right]$
$=-\left(x+y\right)^{2}\left[x^{2}-xy+y^{2}\right]^{2}=-\left(x^{3}+y^{3}\right)^{2}$