if `A_(1) ,B_(1),C_(1) …….` are respectively the cofactors of the elements `a_(1) ,b_(1),c_(1)……` of the determinant
`Delta = |{:(a_(1),,b_(1),,c_(1)),(a_(2),,b_(2),,c_(2)),(a_(3),,b_(3),,c_(3)):}|, Delta ne 0 ` then the value of `|{:(B_(2),,C_(2)),(B_(3),,C_(3)):}|` is equal to

Options

A. `a_(1)^(2)Delta`
B. `a_(1)Delta`
C. `a_(1)Delta^(2)`
D. `a_(1)^(2)Delta^(2)`

Answer

B

Explanation

`B_(2) =a_(1)c_(3) -a_(3)c_(1),C_(2)=-(a_(1)b_(3)-a_(3)b_(1))`
`B_(3) =-(a_(1)C_(2) -a_(2)c_(1)) ,C_(3)=a_(1)b_(2)-a_(2)b_(1)`
` :. |{:(B_(2),,C_(2)),(B_(3),,C_(2)):}|= |{:(a_(1)C_(3)-a_(3)c_(1),,-a_(1)b_(3)+a_(3)b_(1)),(-a_(1)c_(2)+a_(2)c_(1),,a_(1)b_(1)-a_(2)b_(1)):}|`
`=|{:(a_(1)c_(3),,-a_(1)b_(3)),(-a_(1)c_(2),,a_(1)b_(2)):}|+ |{:(a_(1)C_(3),,a_(3)b_(1)),(-a_(1)c_(2),,-a_(2)b_(1)):}|`
`+|{:(-a_(3)C_(1),,-a_(1)b_(3)),(-a_(1)C_(2),,a_(1)b_(2)):}|+ |{:(-a_(3)C_(1),,a_(3)b_(1)),(a_(2)c_(1),,-a_(2)b_(1)):}|`
` =a_(1)^(2) |{:(C_(3),,-b_(3)),(-c_(2),,b_(2)):}|+a_(1)b_(1) |{:(c_(3),,a_(3)),(-c_(2),,-a_(2)):}|`
`+a_(1)c_(1) |{:(-a_(3),,-b_(3)),(a_(2),,b_(2)):}|+b_(1)c_(1) |{:(-a_(3),,a_(3)),(a_(2),,-a_(2)):}|`
`=a_(1){a_(1)(b_(2)c_(3)-b_(3)c_(2))-b_(1)(a_(2)c_(3)-a_(3)c_(2))`
`+c_(1)(a_(2)b_(3)-a_(3)b_(2))}`
` =a_(1)|{:(a_(1),,b_(1),,c_(1)),(a_(2),,b_(2),,c_(2)),(a_(3),,b_(3),,c_(3)):}| =a_(1) Delta`