If `A=[(1,0),(1,1)] and I=[(1,0),(0,1)]` then which one of the following holds for all `nge1` by the principle of mathematica induction? (A) `A^n=2^(n-1) A+(n-1)I` (B) `A^n=nA+(n-1) I` (C) `A^n=2^(n-1) A-(n-1)I` (D) `A^n=nA-(n-1) AI`

Answer

C

Explanation

`A^(2) = [[1,0],[1,1]] [[1,0],[1,1]] =[[1,0],[2,1]]`
`A^(3) = [[1,0],[2,1]] [[1,0],[1,1]] =[[1,0],[3,1]]`
`A^(n) = [[1,0],[n,1]]`
`nA = [[n,0],[n,n]], (n-1) I = [[n-1,0],[0,n-1]]`
`nA-(n-1)I= [[1,0],[n,1]]=A^(n)`