Let A be m m matrix with all elements equal to 1 such that A^n=16¹⁷ A, m, n N , find sum of possible values of n .

Step-by-step solution

If m=2A= [ array ll 1 & 1 1 & 1 array ] , then A^2= [ array ll 2 & 2 2 & 2 array ], A^3= [ array ll 4 & 4 4 & 4 array ], . ., A^n= [ array ll 2^ n-1 & 2^ n-1 2^ n-1 & 2^ n-1 array ] If m=3 , then A^n= [ array lll 3^ n-1 & 3^ n-1 & 3^ n-1 3^ n-1 & 3^ n-1 & 3^ n-1 3^ n-1 & 3^ n-1 & 3^ n-1 array ] If m=m , then A^n= [ array llll m^ n-1 & m^ n-1 & m^ n-1 m^ n-1 & m^ n-1 & & m^ n-1 m^ n-1 & m^ n-1 & & m^ n-1 array ] A^n=m^ n-1 A=16¹⁷ A=2⁶⁸ A Factors of 68 are 1, 2, 4, 17, 34, 68. If n-1=1 n=2 , then m=2⁶⁸ If n-1=2 n=3 ,