If I is the identity matrix of order 2 and A= [ array ll 1 & 1 0 & 1 array ] , then for n 1 , mathematical induction gives

Step-by-step solution

Given A= [ array ll 1 & 1 0 & 1 array ] Now, A^2= [ array ll 1 & 1 0 & 1 array ] [ array ll 1 & 1 0 & 1 array ] = [ array ll 1+0 & 1+1 0+0 & 0+1 array ]= [ array ll 1 & 2 0 & 1 array ] Similarly, A^3= [ array ll 1 & 3 0 & 1 array ] A^n= [ array ll 1 & n 0 & 1 array ] We have, n A-(n-1) I aligned & = [ array ll n & n 0 & n array ]- [ array cc n-1 & 0 0 & n-1 array ] & = [ array ll 1 & n 0 & 1 array ]=A^n A^n & =n A-(n-1) / is true aligned