Let `A=[[0,1],[0,0]]`, prove that: `(aI+bA)^n = a^(n-1) bA`, where I is the unit matrix of order 2 and n is a positive integer.

Let A = [[0,1],[0,0]] , show that (aI + bA)^n = a^nI + na^(n-1) bA , where I is the identity matrix of order 2 and n in N

Show that (-sqrt(-1))^(4n+3)=i , where n is a positive integer.

If A is an idempotent matrix satisfying (I-0.4A)^(-1)=I-alphaA where I is the unit matrix of the same order as that of A then the value of alpha is

Prove that (n!)^2 le n^n. (n!)<(2n)! for all positive integers n.

Prove that the function f(x) = x^n , is continuous at x=n , where n is a positive integer.

If A is any square matrix, prove that (A^n)' = (A')^n , where n is any positive integer.

Using Permutation or otherwise, prove that ((n^2)!)/((n!)^2) is an integer, where n is a positive integer.