Which of the following has`//`have value equal to zero ?

Options

A. `|{:(8,,2,,7),(12,,3,,5),(16,,4,,3):}|`
B. `|{:(1//a,,a^(2),,bc),(1//b,,b^(2),,ac),(1//c,,c^(2),,ab):}|`
C. `|{:(a+b,,2a+b,,3a+b),(2a+b,,3a+b,,4a+b),(4a+b,,5a+b,,6a+b):}|`
D. `|{:(2,,43,,6),(7,,35,,4),(3,,17,,2):}|`

Answer

A, B, C

Explanation

`|{:(8,,2,,7),(12,,3,,5),(16,,4,,3):}|=|{:(8,,2,,7),(4,,1,,-2),(4,,1,,-2):}|" " underset(R_(2) to R_(2)-R_(1))(R_(3) to R^(3) -R_(2)" and ")`
`|{:(1//a,,a^(2),,bc),(1//b,,b^(2),,ac),(1//c,,c^(2),,ab):}|=(1)/(abc) |{:(1,,a^(3),,abc),(1,,b^(3),,abc),(1,,c^(3),,abc):}|`
`[R_(1) to aR_(1),R_(2) to bR_(2) ,R_(3) to cR_(3)]`
`=(abc)/(abc) |{:(1,,a^(3),,1),(1,,b^(3),,1),(1,,c^(3),,1):}|`
[taking abc common from `C_(3)`]
`|{:(a+b,,2a+b,,3a+b),(2a+b,,3a+b,,4a+b),(4a+b,,5a+b,,6a+b):}|=|{:(a+b,,2a+b,,3a+b),(a,,a,,a),(2a,,2a,,2a):}|`
`[R_(3) to R_(3) -R_(2),R_(2) to R_(2)-R_(1)]`
`=0`
`|{:(2,,43,,6),(7,,35,,4),(3,,17,,2):}|=|{:(2,,1,,6),(7,,7,,4),(3,,3,,2):}|" "[C_(2) to C_(2) -7 C_(3)]`
`=|{:(1,,1,,6),(0,,7,,4),(0,,3,,2):}|" "[C_(2) to C_(1) -C_(2)]`
`=2`