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Matrix powers are computed by repeated squaring, needing only about log₂ k multiplications. The Fibonacci matrix [[1,1],[1,0]] raised to k has Fibonacci numbers as its entries.
Determinant of a power
det(Aᵏ) = (det A)ᵏ
Fibonacci matrix
[[1,1],[1,0]]ᵏ = [[F(k+1), F(k)], [F(k), F(k−1)]]
[[89, 55], [55, 34]] — the Fibonacci numbers F₁₁, F₁₀ and F₉.
By the same convention as x⁰ = 1: the empty product of matrices is the multiplicative identity.