Question

Linear Algebra Question#1

y Suurěd lower Q1. Suppose that we have the twin recurrences;, bn+1 62b-48cn and cnt8b,-6m Let bo: 26 and co 16. (a) Diagonalise the underlying matrix and hence find the formulas for bk and ctk for any integer k 20. (b) Use logarithms to predict which is the smallest value of k such that both bu and ck are greater than 1,000,000 and check your answer. 2 -2 1 3 y 3 -3 1 Q2. (a) Carefully use Row Operations to find the inverse of G := [6] -2 . (b) Evaluate the determinant of your answer for (a) by using a co-factor expansion 12 and indicate whether or not you think your answers are correct. (c) Without using diagonlisation, calculate G2 and get its determinant too. Does the determinant fit in with your answer to (b)? (4 (a) Verify that the roots of z3 -39a+ 70 are 2, 5 and -7, using substitution. [1] (b) Give the eigenvectors underlying the recurrence hnt 39hn-1-70hn-2 and Q3.

Answer 1

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1 to a 2 ain A+21) 64 4 420 So 4 432 4-3 5 42-54 30 APD P Thus 5

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f o 5.9 to /0 ne 2n-1 2 -I 1000 UTU