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Let P=[(1,0,0),(3,1,0),(9,3,1)] and Q = ...

Let `P=[(1,0,0),(3,1,0),(9,3,1)]` and Q = `[q_(ij)]` be two `3xx3` matrices such that `Q - P^(5) = I_(3)`. Then `(q_(21)+q_(31))/(q_(32))` is equal to

A

52

B

103

C

201

D

205

Text Solution

Verified by Experts

The correct Answer is:
B
We have, `{:P=[(1,0,0),(4,1,0),(16,4,1)]:}`.
`:. {:P^2=PP=[(1,0,0),(4,1,0),(16,4,1)][(1,0,0),(4,1,0),(16,4,1)]=[(1,0,0),(8,1,0),(16+32,8,1)]:}`
`:. {:P^3=P^2P=[(1,0,0),(8,1,0),(16+32,8,1)][(1,0,0),(4,1,0),(16,4,1)]=[(1,0,0),(12,1,0),(16+32+48,12,1)]:}`
By observing the symmetry, we obtain
`{:P^50=[(1,0,0),(4xx50,1,0),(16+32+48+...50"terms",4xx50,1)]:}`
`rArr{:P^50[(1,0,0),(200,1,0),((16xx50xx51)/2,200,1)]:}`
`:.P^50-Q=I`
`rArr{:Q=P^50-I[(0,0,0),(200,0,0),(20400,200,0)]:}`
`rArr q_21=200,q_31=20400and q_32=200`
`:. (q_31+q_32)/q_21=(20400+200)/200=20600/200=103`
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