Question

If A=[1 0] I=[1 0]

[1 1] and [0 1], then which of the following holds for all n ∈ N?

A. Aⁿ=nA−(n−1)I
B. Aⁿ=2ⁿ⁻¹A−(n−1)I
C. Aⁿ=nA+(n−1)I
D. Aⁿ=2ⁿ⁻¹A+(n−1)I

Answer 1

To determine which of the options holds for all n ∈ N, we can use the properties of matrix exponentiation. The correct option is D. A^n=2^(n-1)A+(n-1)I.

Step-by-step explanation:

To determine which of the options holds for all n ∈ N, we can use the properties of matrix exponentiation. Let's evaluate A^n and compare it with the options given.

A = [1 0] and I = [1 0; 1 1].

A^n = I.A^{n-1} = I.I...(n times) = I^n.A^0 = I^n.I = I^{n+1}

So, the correct option is D. A^n=2^(n-1)A+(n-1)I.