Question

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

• A. $0$
• B. $(x+y+z{)}^3$
• C. $2(x+y+z{)}^3$
• D. $2(x+y+z{)}^2$

Answer 1

Answer: (A)

Let $A=\left[\begin{array}{ccc}x+y& y+z& z+x\\ x& y& z\\ x-y& y-z& z-x\end{array}\right]$
Applying $C_1\to C_1+C_2+C_3$
$=\left[\begin{array}{ccc}2\left(x+y+z\right)& y+z& z+x\\ x+y+z& y& z\\ 0& y-z& z-x\end{array}\right]$
$=\left(x+y+z\right)\left[\begin{array}{ccc}2& y+z& z+x\\ 1& y& z\\ 0& y-z& z-x\end{array}\right]$
Applying $R_2\to 2R_2-R_1$
$=\left(x+y+z\right)\left[\begin{array}{ccc}2& y+z& z+x\\ 0& y-z& z-x\\ 0& y-z& z-x\end{array}\right]$
$=0$ ($\because$ Two rows are identical)