`|[1/c,1/c,-(a+b)/c^2],[-(b+c)/c^2,1/a,1/a],[(-b(b+c))/(a^2c),(a+2b+c)/(ac),(-b(a+b))/(ac^2)]|` is

Options

A. dependent on `a,b,c`
B. dependent on `a`
C. dependent on `b`
D. independent on `a,b` and `c`

Answer

D

Explanation

`(a)` Multiplying `C_(1)` by `a`, `C_(2)` by `b` and `C_(3)` by `c`, we obtain
`Delta=(1)/(abc)|{:((a)/(c),(b)/(c),-(a+b)/(c)),(-(b+c)/(a),(b)/(a),(c)/(a)),(-(b(b+c))/(ac),(b(a+2b+c))/(ac),-(b(a+b))/(ac)):}|`
Applying `C_(1)toC_(1)+C_(2)+C_(3)` we get
`Delta=(1)/(abc)|{:(0,(b)/(c),-(a+b)/(c)),(0,(b)/(a),(c)/(a)),(0,(b(a+2b+c))/(ac),-(b(a+b))/(ac)):}|`
This shows that `Delta` is independent of `a`, `b` and `c`.