Question

The determinant $\left|\begin{array}{ccc}{x}^2& (y+z{)}^2& yz\\ y^2& (z+x{)}^2& zx\\ z^2& (x+y{)}^2& xy\end{array}\right|$ is divisible by:

• A. $x^2+y^2+z^2$
• B. $x-y$
• C. $x-y-z$
• D. $x+y+z$

Answer 1

Answer: (A), (B), (D)

Let $\Delta =\left|\begin{array}{ccc}{x}^2& (y+z{)}^2& yz\\ y^2& (z+x{)}^2& zx\\ z^2& (x+y{)}^2& xy\end{array}\right|$
$C_2\to C_2-2C_3$
$\Delta =\left|\begin{array}{ccc}{x}^2& y^2+z^2& yz\\ y^2& z^2+x^2& zx\\ z^2& x^2+y^2& xy\end{array}\right|$
$\Delta =\left(x^2+y^2+z^2\right)(x-y)(y-z)(z-x)(x+y+z)$