Home
Class 12
MATHS
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.
Promotional Banner

Similar Questions

Explore conceptually related problems

Let A be a ral matrix such that A^(67) = A^(-1) , then :

Let 1 gm s^-1 = alpha Ns then value of alpha is

If alpha is a complex number such that alpha^(2)+alpha+1=0 then alpha^(31) is

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

Suppose A and B be two ono-singular matrices such that AB= BA^(m), B^(n) = I and A^(p) = I , where I is an identity matrix. If m = 2 and n = 5 then p equals to

Suppose A and B be two ono-singular matrices such that AB= BA^(m), B^(n) = I and A^(p) = I , where I is an identity matrix. The relation between m, n and p, is

If A is a square matrix such that |A| = 2 , then for any + ve integar n , |A^(n)| is equal to :

If alpha is a complex number such that alpha^(2) - alpha + 1 = 0 , then alpha^(2011) =

Suppose A and B be two ono-singular matrices such that AB= BA^(m), B^(n) = I and A^(p) = I , where I is an identity matrix. Which of the following orderd triplet (m, n, p) is false?