If `A=[(a, b),(0, 1)]` , prove that `A^n=[(a^n ,b((a^n-1)/(a-1))),(0, 1)]` for every positive integer `n` .

Explanation

Given that,

`A= [(a, b),(0, 1)]`

We have to prove that `A^n=[(a^n ,b((a^n-1)/(a-1))),(0, 1)]` for all positive integers `ndot`

It is true for `n=1`.

`implies A^2= [(a ,b),( 0, 1)]xx[(a ,b),( 0, 1)]`

`implies A^2= [(a^2,ab+b),(0,1)]= [(a^2,b(a+1)),(0,1)] = [(a^2,b({a^2-1}/{a-1})),(0,1)]`

It is true for `n=2`.

Let it is true for `n=n-1`

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