Question

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

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

Answer 1

Answer: (B)

$2(x+y+z{)}^3$

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

$C_1\to C_1+C_2+C_3$

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

$\Rightarrow 2(x+y+z)\left|\begin{array}{ccc}1& x& y\\ 1& 2x+y+z& y\\ 1& x& x+2y+z\end{array}\right|$

$R_2\to R_2-R_1,R_3\to R_3-R_1$

$\Rightarrow 2(x+y+z)\left|\begin{array}{ccc}1& x& y\\ 0& (x+y+z)& 0\\ 0& 0& x+y+z\end{array}\right|$

$\Rightarrow 2(x+y+z)[{(x+y+z)}^2-0+0]$

$=2{(x+y+z)}^3$