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Linear Algebra Solutions 12

Linear Algebra Solutions 12 - which is equivalent to the...

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Unformatted text preview: which is equivalent to the stated result. 9. In terms of matrices, we have Fn+1 Fn Fn+1 Fn 11 10 = n 11 10 = Fn Fn−1 F1 F0 for n ≥ 1. n 11 10 = 1 0 . Now λ1 , λ2 are√ roots of the polynomial x2 − x − 1 here. the √ 1+ 5 Hence λ1 = 2 and λ2 = 1−2 5 and √ n−1 √ n−1 1+ 5 − 1−2 5 2 √ √ 1+ 5 − 1−2 5 2 √ n−1 √ n−1 1+ 5 − 1−2 5 2 kn = √ = 5 . Hence An = kn A − λ1 λ2 kn−1 I2 = kn A + kn−1 I2 So Fn+1 Fn = (kn A + kn−1 I2 ) 1 0 1 1 1 0 = kn Hence Fn = k n = + kn−1 √ n−1 1+ 5 2 − √ 5 kn + kn−1 kn = √ n−1 1− 5 2 10. From Question 5, we know that xn yn = 1r 11 15 n a b . . . ...
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