MIT18_06S10_pset7_s10_soln

MIT18_06S10_pset7_s10_soln - 18.06 Problem Set 7 Solutions...

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Unformatted text preview: 18.06 Problem Set 7 Solutions Total: 100 points Prob. 16, Sec. 5.2, Pg. 265: F n is the determinant of the 1 , 1 , − 1 tridiagonal matrix of order n : 1 1 − 1 1 1 − 1 1 1 − 1 1 − 1 1 = 2 1 − 1 1 = 3 = 4 . F 2 F 3 F 4 = = = 1 1 − 1 1 1 1 Expand in cofactors to show that F n = F n − 1 + F n − 2 . These determinants are Fibonacci numbers 1 , 2 , 3 , 5 , 8 , 13 , . . . . The sequence usually starts 1 , 1 , 2 , 3 (with two 1’s) so our F n is the usual F n +1 . Solution (see pg. 535, 4 pts.): The 1 , 1 cofactor of the n by n matrix is F n − 1 . The 1 , 2 cofactor has a 1 in column 1, with cofactor F n − 2 . Multiply by ( − 1) 1+2 and also ( − 1) from the 1 , 2 entry to find F n = F n − 1 + F n − 2 (so these determinants are Fibonacci numbers). Prob. 32, Sec. 5.2, Pg. 268: Cofactors of the 1 , 3 , 1 matrices in Problem 21 give a recursion S n = 3 S n − 1 − S n − 2 . Amazingly that recursion produces every second Fibonacci number. Here is the challenge. Show that S n is the Fibonacci number F 2 n +2 by proving F 2 n +2 = 3 F 2 n − F 2 n − 2 . Keep using Fibo- nacci’s rule F k = F k − 1 + F k − 2 starting with k = 2 n + 2. Solution (see pg. 535, 12 pts.): To show that F 2 n +2 = 3 F 2 n − F 2 n − 2 , keep using Fibonacci’s rule: F 2 n +2 = F 2 n +1 + F 2 n = F 2 n + F n − 1 + F 2 n = 2 F 2 n + ( F 2 n − F 2 n − 2 ) = 3 F 2 n − F 2 n − 2 . Prob. 33, Sec. 5.2, Pg. 268: The symmetric Pascal matrices have determinant 1. If I subtract 1 from the n, n entry, why does the determinant become zero? (Use rule 3 or cofactors.) 1 1 1 1 1 1 1 1 det 1 2 3 4 1 3 6 10 = 1 (known) det 1 2 3 4 1 3 6 10 = (to explain) . 1 4 10 20 1 4 10 19 Solution (see pg. 535, 12 pts.): The difference from 20 to 19 multiplies its cofactor, which is the determinant of the 3 by 3 Pascal matrix, so equal to 1. Thus the det drops by 1....
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This note was uploaded on 08/22/2011 for the course MATH 1806 taught by Professor Strang during the Fall '10 term at MIT.

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MIT18_06S10_pset7_s10_soln - 18.06 Problem Set 7 Solutions...

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