Home
Class 12
MATHS
If A = [[1 ,1],[1,1]] and det (A^(n) - ...

If `A = [[1 ,1],[1,1]]` and det `(A^(n) - 1) = 1 -lambda ^(n), n in N,` then the value of `lambda` is

A

1

B

2

C

3

D

4

Text Solution

Verified by Experts

The correct Answer is:
B
`because " "A = [[1 ,1],[1,1]]`
`therefore " "A^(2) = [[2 ,2],[2,2]] = 2[[1 ,1],[1,1]] =2A`
` rArr A^(3) = A^(2) cdot A = 2A^(2) = 2^(2) A`
Similarly, `A_(n) = 2^(n-1) A `
`therefore " "A^(n) -I= [[2^(n-1) ,2^(n-1)],[2^(n-1),2^(n-1)]] -[[1 ,0],[0,1]]`
`= [[2^(n-1)-1 ,2^(n-1)],[2^(n-1),2^(n-1)-1]] `
`rArr "det" (A^(n)-I) = (2^(n-1)-1)^(2) - (2^(n-1))^(2)`
`= 1- 2^(n) = 1 - lambda ^(n)` [given]
`therefore lambda = 2`
Promotional Banner

Similar Questions

Explore conceptually related problems

If A=[(1,1), (1,1)] and det (A^n-1)=1-lambda^n, n in NN, then the value of lambda_n is

If A=det[[1,11,1]] and A^(100)=lambda^(99)A, then the value of lambda is

If ^(2n+1)P_(n-1):^(2n-1)P_(n)=3:5, then find the value of n.

If y=ax^(n+1)+bx^(-n) and x^(2)(d^(2)y)/(dx^(2))=lambda y then write the value of lambda.

" If "A=[[lambda,2,-1],[-1,1,2],[2,-1,1]]" and "det(adj(adj A)=(23)^(4)," then one of the possible value of "lambda(lambda>0)can be

If P(n):2xx4^(2n+1)=3^(3n+1) is divisible by lambda for all nN is true,then find the value of lambda

Let A be an n xx n matrix. If det (lambda A) = lambda^(s) det(A), what is the value of s?

If x^(n)-1 is divisible by x-lambda, then the least prositive integral value of lambda is 1 b.3 c.4 d.2