Question

$\left|\begin{array}{ccc}y+z& x& x\\ y& z+x& y\\ z& z& x+y\end{array}\right|$ equals

• A. $x^2{y}^2{z}^2$
• B. $4x^2{y}^2{z}^2$
• C. $xyz$
• D. $4xyz$

Answer 1

The correct option is D $4xyz$

Let $\Delta =\left|\begin{array}{ccc}y+z& x& x\\ y& z+x& y\\ z& z& x+y\end{array}\right|$
Applying $\Delta =\left|\begin{array}{ccc}y+z& x& x\\ y& z+x& y\\ z& z& x+y\end{array}\right|$
$\Delta =\left|\begin{array}{ccc}y+z& x& x\\ -z& z& y-x\\ -y& z-x& y\end{array}\right|=(y+z)(yz-(y-x)(z-x))-x(-yz+y(y-x))+x(yz+-z(z-x))=(y+z)(yz-yz+yx+xz-x^2)-x(-yz-yx+y^2)+x(xz+yz+{-z}^2)=y^{2}x+y^{2}x-x^{2}y+y^{2}x+z^{2}x-x^{2}z+y^{2}x-y^{2}x+x^{2}y-z^{2}x+x^{2}z+y^{2}x=4xyz$