Let K be a positive real number and `A=[(2k-1,2sqrt(k),2sqrt(k)),(2sqrt(k),1,-2k),(-2sqrt(k),2k,-1)]` and `B=[(0,2k-1,sqrt(k)),(1-2k,0,2sqrt(k)),(-sqrt(k),-2sqrt(k),0)]`. If det (adj A) + det (adj B) `=10^(6)`, then `[k]` is equal to ______ .
[Note : adj M denotes the adjoint of a square matrix M and `[k]` denotes the largest integer less than or equal to `k`.]

Answer

4

Explanation

`|A|=(2k+1)^(3), |B|=0` (since B is a skew-symmetric matrix of order 3)
`implies` det (adj A)`=|A|^(n-1)=((2k+1)^(3))^(2)=10^(6)`
`implies 2k+1=10` or `2k=9`
`implies [K]=4`.