Question

Let `A = [[1,0,0],[1,0,1], [0,1,0]] " satisfies " A^(n) = A^(n-2) + A^(2 ) -I` for `nge 3` and
consider matrix `underset(3xx3)(U)` with its columns as `U_(1), U_(2), U_(3),` such that
`A^(50)U_(1)=[[1],[25],[25]],A^(50) U_(2)=[[0],[1],[0]]and A^(50) U_(3)[[0],[0],[1]]`
Trace of `A^(50)` equals

Answer 1

Correct Answer - D
`because A^(n) = A^(n-2) + A^(2) - I rArr A^(50) = A^(48) + A^(2) - I`
Further, `{:(A^(48) =, A^(46) +, A^(2) ,- I) , (A^(46) =, A^(44) +, A^(2), -I),(vdots ,vdots,vdots, vdots),(A^(4)=,A^(2) +, A^(2),-I^(4)):} `
On adding all, we get
` A^(50) = 25 A^(2) - 24I` ...(i)
`because A^(2) = [[1,0,0],[1,0,1],[0,1,0]][[1,0,0],[1,0,1],[0,1,0]]= [[1,0,0],[1,1,0],[1,0,1]]`
`therefore A^(50) = 25 A^(2) - 24I = [[25,0,0],[25,25,0],[25,0,25]]-[[24,0,0],[0,24,0],[0,0,24]]` [from Eq. (i)]
`=[[1,0,0],[25,1,0],[25, 0,1]]` ...(ii)
Hence, trace of `A^(50) = 1+ 1+ 1= 3`