Question

Evaluate the value of :

$\left|\begin{array}{ccc}x& 1& y+z\\ y& 1& z+x\\ z& 1& x+y\end{array}\right|$ =

• A. $1+x+y+z$
• B. $x+y+z$
• C. $0$
• D. $1$

Answer 1

Answer: (C)

$0$

$A=\left|\begin{array}{ccc}x& 1& y+z\\ y& 1& z+x\\ z& 1& x+y\end{array}\right|$
By column trasformation
$C_3\to C_3+C_1$
$\Rightarrow \left|\begin{array}{ccc}x& 1& x+y+z\\ y& 1& x+y+z\\ z& 1& x+y+z\end{array}\right|$
take common $(x+y+z)$
$\Rightarrow (x+y+z)\left|\begin{array}{ccc}x& 1& 1\\ y& 1& 1\\ z& 1& 1\end{array}\right|$
So two rows are same
So, $A=0$