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= [[1,0,0],[4,1,0],[16,4,1]]= I + [[0,0,0],[4,0,0],[16,4,0]]= I + A `
Let `A=[[0,0,0],[4,0,0],[16,4,0]] `
`rArr A^(2)=[[0,0,0],[0,0,0],[16,0,0]]and A^(3) = [[0,0,0],[0,0,0],[0,0,0]] `
`rArr A^(n)` is a null matrix `Aan ge 3`
`therefore P^(50) = (I+A) ^(50) = I + 50 A + (50xx49)/2 A^(20)`
`rArr Q + j= I +50A + 25xx49 A^(2)`
or `Q = 50 A + 25 xx 49 A^(2)`
`=[[0,0,0],[200,0,0],[800,200,0]]+ [[0,0,0],[0,0,0],[19600,0,0]] `
`therefore = [[q_(11),q_(12),q_(13)],[q_(21),q_(22),q_(23)],[q_(31),q_(32),q_(33)]]+[[0,0,0],[200,0,0],[20400,200,0]] `
On comparing, we get
`q_(11) = q_(32) = 200 , q_(31) = 20400 `
`therefore (q_(31)+q_(32))/q_(21) = (20400+ 200)/200`
`= 102 + 1 = 103`