If `D_k=|[1, n, n];[ 2k, n^2+n+1, n^2+n];[ 2k-1, n^2, n^2+n+1]|a n dsum_(k=1)^n D_k=56.` then `n` equals
`(a) 4`
(b)`6`
(c)`8`
(d) none of these

Options

A. 4
B. 6
C. 8
D. 7

Explanation

`overset(n)underset(k=1)(Sigma) D_(k) =56`
`rArr |{:(overset(n)underset(k=1)(Sigma)1,,n,,n),(overset(n)underset(k=1)(Sigma)2k,,n^(2)+n+1,,n^(2)+n),(overset(n)underset(k=1)(Sigma)(2k-1),,n^(2),,n^(2)+n+1):}|=56`
`" or " |{:(n,,n,,n),(n(n+1),,n^(2)+n+1,,n^(2)+n),(n^(2),,n^(2),,n^(2)+n+1):}|`
Applying `C_(3) to C_(3) -C_(1) " and " C_(2) to C_(2)-C_(1)` we get
`|{:(n,,0,,0),(n(n+1),,1,,0),(n^(2),,0,,n+1):}|=56`
`" or " n(n+1) =56 " or " n=7`