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Consider three matrices A=[(2,1),(4,1)],...

Consider three matrices `A=[(2,1),(4,1)], B=[(3,4),(2,3)] and C=[(3,-4),(-2,3)].` Then the value of the sum `tr(A)+tr((ABC)/2) +tr((A(BC)^2)/4)+tr((A(BC)^3)/8)+.......+oo` is (A) `6` (B) `9` (C) `12` (D) none of these

A

4

B

9

C

12

D

6

Text Solution

Verified by Experts

The correct Answer is:
D
`because BC = [[3,4],[2,3]] [[3,-4],[-2,3]]=[[1,0],[0,1]]=I`
`therefore tr (A) + tr ((ABC)/2) + tr ((A(BC)^(2))/4) +tr((A(BC)^(3))/8) +...`
`= tr (A) + tr (A/2) + tr(A/2^(2))+tr(A/2^(3))+..." upto " infty`
`=tr (A) + 1/2 tr (A) + 1/2^(2) tr (A) + ... " upt0 " infty `
`= (tr(A))/(1-(1/2)) = 2 tr (A) = 2(2+1) = 6`
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