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Let A be a nxxn matrix such thatA ^(n) =...

Let A be a `nxxn` matrix such that`A ^(n) = alpha A,` where `alpha ` is a
real number different from 1 and - 1. The matrix `A + I_(n)` is

A

singular

B

invertible

C

scalar matrix

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
B
Let `B=A + I_(n)`
`therefore A= B-I_(n)`
Given, `A^(n) =alpha A `
`rArr (B-I_(n) ) ^(n) = alpha (B-I_(n))`
`rArr B^(n) -""^(n)C_(1) B^(n-1) + ""^(n) C_(2) B^(n-2) +...+ (-1) ^(n) I_(n)`
`= alpha B - alphaI_(n)`
`rArr B( B^(n-1) -""^(n)C_(1) B^(n-2) + ""^(n) C_(2) B^(n-3) +...+ (-1) ^(n-1) I_(n)-alphaI_(n))`
`= [(-1)^(n+1) - alpha ]I_(n ) ne 0 [ because alpha ne pm 1]`
Hence, B is invertible.
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