Let B be the 2x2 matrix [6 -5 2 -1]. If 2x1 vector v = [2 1]. Find the formula for (A^n)(v) in terms of n.

Answer 1

Need to use A^{n =} PDⁿP^-1 and then multiply Aⁿv.

Turns out D = [4 0;0 1], P = [5 1;2 1] and P^-1= [1/3 -1/3;-2/3 5/3]

Dⁿ = [4ⁿ 0;0 1]

so just do the matrix-vector multiplication

I got [(5/3)4ⁿ+1/3; (2)4ⁿ+1/3]