Question

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

• A. $(x+y+z)(x-z{)}^2$
• B. xyz
• C. 2(x+y+z)
• D. xyz$(x-z{)}^2$

Answer 1

The correct option is A $(x+y+z)(x-z{)}^2$

Performing $R_1\to R_1+R_2+R_3$

$(x+y+z)\left|\begin{array}{ccc}z& 1& 1\\ z+x& z& x\\ x+y& y& z\end{array}\right|$
Performing $C_1\to C_1+C_2+C_3$
$(x+y+z)\left|\begin{array}{ccc}0& 1& 1\\ 0& z& x\\ x-z& y& z\end{array}\right|$
$=(x+y+z)(x-z{)}^2$