If A = [[1, 1], [0, 1]] and n is a positive integer, then Aⁿ is equal to:

If A = [[1, 1], [0, 1]] and n is a positive integer, then Aⁿ is equal to:

• A. [[1, n], [0, 1]]
• B. [[1, 0], [0, 1]]
• C. [[n, n], [0, n]]
• D. [[1, 1], [0, 1]]

Answer: A) [[1, n], [0, 1]]

Explanation: This can be proved by induction. A¹ = [[1, 1], [0, 1]]. Assume Aᵏ = [[1, k], [0, 1]]. Then Aᵏ⁺¹ = Aᵏ A = [[1, k], [0, 1]] × [[1, 1], [0, 1]] = [[1×1+k×0, 1×1+k×1], [0×1+1×0, 0×1+1×1]] = [[1, 1+k], [0, 1]]. So Aⁿ = [[1, n], [0, 1]].