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If A = [1 0; 1 1] and I = [1 0; 0 1], then which one of the following holds for all n ≥ 1, by the principle of mathematical induction

1. Aⁿ = nA − (n − 1)I
2. Aⁿ = 2ⁿ − 1 A − (n − 1)I
3. Aⁿ = nA + (n − 1)I
4. Aⁿ = 2ⁿ − 1 A + (n − 1)I

Correct answer: Aⁿ = nA − (n − 1)I

Solution

The correct option is valid because it can be proven using mathematical induction. By establishing a base case and then assuming it holds for n, we can show that it also holds for n+1, confirming that the relationship between Aⁿ, nA, and I is consistent for all n ≥ 1.

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