If \(A = \begin{pmatrix} 1 & 2 \\ 2 & -1 \end{pmatrix} \) ,then which ONE of…

Key step: compute A^2 and use powers of that result.

Start with the matrix A = [[1, 2], [2, -1]].

Compute A^2 by multiplying A with itself:

• A^2 = [[1*1 + 2*2, 1*2 + 2*(-1)], [2*1 + (-1)*2, 2*2 + (-1)*(-1)]] = [[5, 0], [0, 5]] = 5 I.

• Therefore A^8 = (A^2)^4 = (5 I)^4 = 5^4 I = 625 I.

Final answer: A^8 = [[625, 0], [0, 625]].

Common mistakes:

• [[25, 0], [0, 25]] equals A^4 because it is (A^2)^2 = 5^2 I.

• [[125, 0], [0, 125]] equals A^6 because it is (A^2)^3 = 5^3 I.

• [[3125, 0], [0, 3125]] equals A^10 because it is (A^2)^5 = 5^5 I.