Question

Value of $\left|\begin{array}{ccc}x+y& z& z\\ x& y+z& x\\ y& y& z+x\end{array}\right|$, where $x,y,z$ are nonzero real numbers, is equal to

• A. $xyz$
• B. $2xyz$
• C. $3xyz$
• D. $4xyz$

Answer 1

The correct option is D $4xyz$

$\left|\begin{array}{ccc}x+y& z& z\\ x& y+z& x\\ y& y& z+x\end{array}\right|$
$R_1\to R_1+R_2+R_3$
$=\left|\begin{array}{ccc}2(x+y)& 2(y+z)& 2(x+z)\\ x& y+z& x\\ y& y& z+x\end{array}\right|$
$=2\left|\begin{array}{ccc}(x+y)& (y+z)& (x+z)\\ x& y+z& x\\ y& y& z+x\end{array}\right|$
$R_1\to R_1-R_2,R_2\to R_2-R_3$
$=2\left|\begin{array}{ccc}y& 0& z\\ x-y& z& -z\\ y& y& z+x\end{array}\right|$
$R_2\to R_2+R_1$
$=2\left|\begin{array}{ccc}y& 0& z\\ x& z& 0\\ y& y& z+x\end{array}\right|$
$=2\left[yz\right(z+x)+z(xy-yz\left)\right]$
$=4xyz$