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Matrix A such that A^2=2A-I ,w h e r eI ...

Matrix `A` such that `A^2=2A-I ,w h e r eI` is the identity matrix, the for `ngeq2. A^n` is equal to

A

`nA-(n-1)l`

B

`nA-l`

C

`2^(n-1)A=(n-1)l`

D

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

Text Solution

Verified by Experts

The correct Answer is:
A
`:'A^(2)=2A-l` [Given]
`:.A^(2)A=(2A-l)A=2"AA"-lA`
`=2A^(2)-A=2(2A-l)-A`
`impliesA^(3)=3A-2l`
`impliesA^(3).A=3"AA"-2lA`
`=3(2A-l)-2A`
`impliesA^(4)=4A-3l`
Similarly, `A^(n)=nA-(n-1)l`
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