Using properties of determinants, prove the following: `|xx+y x+2y\ x+2y xx+y x+y x+2y x|=9y^2(x+y)`

Options

A. `9x^(2)(x+y)`
B. `9y^(2)(x+y)`
C. `3y^(2)(x+y)`
D. `7x^(2)(x+y)`

Answer

B

Explanation

We have `|(x,x+y,x+2y),(x+2y,x,x+y),(x+y,x+2y,x)|`
`=|(3(x+y),x+y,y),(3(x+y),x,y),(3(x+y),x+2y,-2y)| [ :' C_(1)toC_(1)+C_(2)+C_(3)` and `C_(3)toC_(3)-C_(2)]`
`=3(x+y)|(1,(x+y),y),(1,x,y),(1,(x+2y),-2y)|` [taking `3(x+y)` common from first column]
`=3(x+y)|(0,y,0),(1,x,y),(1,(x+2y),-2y)| [ :' R_(1)toR_(1)-R_(2)]`
Expanding along `R_(1)`
`=3(xy)[-y(-2y-y)]`
`=3y^(2).3(x+y)=9y^(2)(x+y)`