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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-1

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-I implies A^(2)*A=(2A-I)A`
`implis A^(#)=2A^(2)-IA=2(2A-I)-A implies A^(3)=3A-2I" "{because IA=A and A^(2)=2A-I}`
Similarly, `A^(4)=4A-3I,A^(5)=5A-4I implies A^(n)=nA-(n-1)I`
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