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Matrix A such that A^(2)=2A-I, where I i...

Matrix A such that `A^(2)=2A-I`, where I is the identity matrix, then for `n ge 2, A^(n)` is equal to

A

`nA-(n-1)I`

B

`nA-I`

C

`2^(n-1)A-(n-1)I`

D

`2^(n-1)A-I`

Text Solution

Verified by Experts

The correct Answer is:
A
`"As we have "A^(2)=2A-IrArrA^(2).A=2A-IA`
`rArr" "A^(3)=2A^(2)-IA=2(2A-I)-A rArr A^(3)=3A-2I" "{because IA=A and A^(2)=2A-I}`
`"Similarly, "A^(4)=4A-3I, A^(5)=5A-4I. rArr a^(n)=nA-(n-1)I`
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