Let `A ,B` be two matrices such that they commute, then for any positive integer `n ,`
(i) `A B^n=B^n A`
(ii)`(A B)^n =A^n B^n`

If A is a nilpotent matrix of index 2, then for any positive integer n ,A(I+A)^n is equal to (a) A^(-1) (b) A (c) A^n (d) I_n

Statement 1: Let A ,B be two square matrices of the same order such that A B=B A ,A^m=O ,n dB^n=O for some positive integers m ,n , then there exists a positive integer r such that (A+B)^r=Odot Statement 2: If A B=B A then (A+B)^r can be expanded as binomial expansion.

Let A, B be two matrices different from identify matrix such that AB=BA and A^(n)-B^(n) is invertible for some positive integer n. If A^(n)-B^(n)=A^(n+1)-B^(n+1)=A^(n+2)-B^(n+2) , then

If A and B are square matrices of the same order such that A B = B A , then prove by induction that A B^n=B^n A .

For any two sets,prove that n(Acup B)=n(A)+n(B)-n(AcapB) .

If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB^(n)=B^(n)A . Further, prove that (AB)^(n)=A^(n)B^(n) for all n in N .

If i=sqrt-1 and n is a positive integer, then i^(n)+i^(n+1)+i^(n+3_ is equal to

If Aa n dB are two matrices such that A B=Ba n dB A=A ,t h e n (A^5-B^5)^3=A-B b. (A^5-B^5)^3=A^3-B^3 c. A-B is idempotent d. none of these

If A and B are two sets such that n(A) = 27, n(B) = 35 and n(A cup B) = 50 , then n (A cap B) =

A and B are square matrices and A is non-singular matrix, then (A^(-1) BA)^n,n in I' ,is equal to (A) A^-nB^nA^n (B) A^nB^nA^-n (C) A^-1B^nA (D) A^-nBA^n