Question

The value of $\left|\begin{array}{ccc}z+y& x& x\\ y& z+x& y\\ z& z& x+y\end{array}\right|$ is equal to

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

Answer 1

Answer: (B)

4xyz

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

$R_1\to R_1-(R_1+R_1)$

$\Rightarrow \left|\begin{array}{ccc}0& -2z& -2y\\ y& x+z& y\\ z& z& x+y\end{array}\right|$

$\Rightarrow -(-2z)(y(x+y)-yz)-2y(yz-z(x+z\left)\right)$

$\Rightarrow 2z(xy+y^2-yz)-2y(yz-xz-z^2)$

$\Rightarrow 2[xyz+y^{2}z-y{z}^2-y^{2}z+xyz+y{z}^{}]$

$\Rightarrow 4xyz$

Option B is correct