The determinant `|{:(a+b+c,a+b,a),(4a+3b+2c,3a+2b,2a),(10a+6b+3c,6a+3b,3a):}|` is independent of which one of the following ?

Options

A. a and b
B. b and c
C. a and c
D. All of these

Answer

B

Explanation

Let `D=|{:(" "a+b+c,a+b,a),(4a+3b+2c,3a+2b,2a),(10a+6b+3c,6a+3b,3a):}|`
By `R_(2) rarr R_(2)-2R_(1)and R_(3)rarr R_(3)-3R_(1)`, we get:
`rArr |{:(a+b+c,a+b,a),(2a+b,a,0),(7a+3b,3a,0):}|`
By `C_(1) rarr C_(2)` gives:
`rArr |{:(" "c,a+b,a),(a+b,a,0),(4a+3b,3a,0):}|`
Again by `R_(3) rarr R_(3)-3R_(2)`, we get:
`D=|{:(" "c,a+b," "a),(a+b," "a,0),(a," "0,0):}|`
`= a{0.(a+b)-a.a}`
`= -a^(3)` which is independent of b and c.