Let A = 0 i i 0 , where i 2 = - 1 . Let I denotes the identity matrix of order 2, then I + A + A 2 + A 3 … … . A 110 is equal to

Question

Let A = 0 i i 0 , where i 2 = - 1 . Let I denotes the identity matrix of order 2, then I + A + A 2 + A 3 … … . A 110 is equal to, 0 i i 0, 0 0 0 0, 1 0 0 1, - 1 0 0 0

MARKS App · Accepted Answer

A 2 = 0 i i 0 0 i i 0 = - 1 0 0 - 1 = - I A 3 = A 2 A = − I ⋅ A = − A A 4 = A 2 2 = I A 5 = A , A 6 = - I , A 7 = - A , A 8 = I … . So on I + A + A 2 + A 3 = I + A - I - A = 0 A 4 + A 5 + A 6 + A 7 = 0 … … … … A 104 + A 105 + A 106 + A 107 = 0 A 108 + A 109 + A 110 = I + A - I = A ⇒ I + A + A 2 + . … … . + A 110 = A

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