If a, b, c are real numbers, then the value the determinant ` |(1-a,a-b-c,b+c),(1-b,b-c-a,c+a),(1-c,c-a-b,a+b)|` is

Options

A. 0
B. `(a-b)(b-c)(c-a)`
C. `(a+b+c)^(2)`
D. `(a+b+c)^(3)`

Answer

A

Explanation

`|(1-a,a-b-c,b+c),(1-b,b-c-a,c+a),(1-c,c-a-b,a+b)|`
apply `C_(2)rarrC_(2)rarr+C_(3)`
`|(1-a,a,b+c),(1-b,b,c+a),(1-c,c,a+b)|`
apply `C_(1)rarrC_(1)rarr+C_(2)`
`|(1,a,b+c),(1,b,c+a),(1,c,a+b)|`
apply `C_(3)rarrC_(2)+C_(3)`
`(a+b+c)|(1,a,1),(1,b,1),(1,c,1)|=0`