Let `P=[[1,0,0],[4,1,0],[16,4,1]]`and `I` be the identity matrix of order `3`. If `Q = [q_()ij ]` is a matrix, such that `P^(50)-Q=I`, then `(q_(31)+q_(32))/q_(21)` equals

Options

A. 52
B. 103
C. 201
D. 205

Answer

B

Explanation

`P^(2)=[(1,0,0),(4,1,0),(16,4,1)][(1,0,0),(4,1,0),(16,4,1)]=[(1,0,0),(8,1,0),(48,8,1)]`
`P^(3)=[(1,0,0),(8,1,0),(48,8,1)][(1,0,0),(4,1,0),(16,4,1)]=[(1,0,0),(12,1,0),(96,12,1)]`
`:. P^(n)=[(1,0,0),(4n,1,0),(8(n^(2)+n),4n,1)]`
`:. P^(50)=[(1,0,0),(200,1,0),(8xx50/951),200,1)]`
`P^(50)-Q=I`
`:.` Equating w get
`200-q_(21)=0implies q_(21)=200`
`400xx51-q_(31)=0`
`implies q_(31)=400xx51`
`200-q_(32)=0 implies q_(32)=200`
`implies (q_(31)+q_(32))/q_(21)=(400xx51+200)/(200) =2(51)+1=103`