If A is a matrix [ array ll 1 & 0 1 & 1 array ] , then A^n= [ array ll 1 & 0 n & 1 array ] , n N

Step-by-step solution

Given, If A= [ array ll 1 & 0 1 & 1 array ] , then A^n= [ array ll 1 & 0 n & 1 array ], n N We will generalise the exponent of A . Let n=2 aligned A^2 & = [ array ll 1 & 0 1 & 1 array ] [ array ll 1 & 0 1 & 1 array ] & = [ array cc 1 1+0 1 & 1 0+0 1 1 1+1 1 & 1 0+1 1 array ]= [ array ll 1 & 0 2 & 1 array ] aligned Again, let n=3 A^3=A^2 A= [ array ll 1 & 0 2 & 1 array ] [ array ll 1 & 0 1 & 1 array ]= [ array cc 1 & 0 2+1 & 1 array ]= [ array ll 1 & 0 3 & 1 array ] Similarly, A^4= [ array ll 1 & 0 4 & 1 array ] Hen