If `A=d i ag\ (a\ \ b\ \ c)` , show that `A^n=d i ag\ (a^n\ \ b^n\ \ c^n)` for all positive integer `n` .

If A=d i ag(abc), show that A^n=d i ag(a^nb^nc^n) for all positive integer n .

Using mathematical induction prove that d/(dx)(x^n)=n x^(n-1) for all positive integers n.

If aa n db are distinct integers, prove that a-b is a factor of a^n-b^n , wherever n is a positive integer.

If Aa n dB are square matrices of the same order and A is non-singular, then for a positive integer n ,(A^(-1)B A)^n is equal to A^(-n)B^n A^n b. A^n B^n A^(-n) c. A^(-1)B^n A^ d. n(A^(-1)B^A)^

If A=[0 1 0 0] , prove that (a I+b A)^n=a^n\ I+n a^(n-1)\ b A where I is a unit matrix of order 2 and n is a positive integer.

Let A ,\ B be two matrices such that they commute. Show that for any positive integer n , A B^n=B^n A

Let A ,\ B be two matrices such that they commute. Show that for any positive integer n , (A B)^n=A^n B^n

If A a n d B are two non-singular matrices of the same order such that B^r=I , for some positive integer r >1,t h e n (A^(-1)B^(r-1)A)-(A^(-1)B^(-1)A)= a. I b. 2I c. O d. -I

Let A ,B be two matrices such that they commute. Show that for any positive integer n , A B^n=B^n A

(x+1) is a factor of x^n+1 only if (a)n is an odd integer (b) n is an even integer (c )n is a negative integer (d)n is a positive integer