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If A = [[0, 1],[3,0]]and (A^(8) + A^(6) ...

If `A = [[0, 1],[3,0]]and (A^(8) + A^(6) + A^(4) + A^(2) + I) V= [[0],[11]],`
where `V` is a vertical vector and `I` is the `2xx2` identity
matrix and if `lambda` is sum of all elements of vertical vector
`V`, the value of `11 lambda` is

Text Solution

Verified by Experts

The correct Answer is:
1
`because A =[[0,1],[3,0]]`
`therefore A^(2) = A cdot A = [[0,1],[3,0]] [[0,1],[3,0]]= [[3,0],[0,3]]= 3I`
`rArr A^(4) = (A^(2))^(2) = 9 I , A^(6) = 27I, A^(8) = 81 I`
Now, `(A^(8) + A^(6) + A^(4)+ A^(2) + I) V = (121) IV = (121) V" "...(i)`
`(A^(8) + A^(6) + A^(4)+ A^(2) + I) V = [[0],[11]]" " (ii) `
From Eqs. (i) and (ii), `(121) V = [[0],[11]]rArr V = [[0],[1/11]]`
`therefore` Sun of elements of `V= 0 + 1/11 = 1/11 = lambda` [given]
`therefore 11 lambda = 1`
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