A 2×22 \times 22×2 matrix AAA has eigenvalues λ1=3,λ2=−1\lambda_1 = 3, \lambda_2 = -1λ1=3,λ2=−1 with eigenvectors v1=(11),v2=(1−1)v_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}, v_2 = \begin{pmatrix} 1 \\ -1 \end{pmatrix}v1=(11),v2=(1−1). What is Ak(20)A^k \begin{pmatrix} 2 \\ 0 \end{pmatrix}Ak(20)?
3k(11)+(−1)k(1−1)3^k \begin{pmatrix} 1 \\ 1 \end{pmatrix} + (-1)^k \begin{pmatrix} 1 \\ -1 \end{pmatrix}3k(11)+(−1)k(1−1)
3k(11)−(−1)k(1−1)3^k \begin{pmatrix} 1 \\ 1 \end{pmatrix} - (-1)^k \begin{pmatrix} 1 \\ -1 \end{pmatrix}3k(11)−(−1)k(1−1)
2⋅3k(11)2 \cdot 3^k \begin{pmatrix} 1 \\ 1 \end{pmatrix}2⋅3k(11)
(3k+(−1)k3k−(−1)k)\begin{pmatrix} 3^k + (-1)^k \\ 3^k - (-1)^k \end{pmatrix}(3k+(−1)k3k−(−1)k)