This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
This note was uploaded on 08/22/2011 for the course MATH 1806 taught by Professor Strang during the Fall '10 term at MIT.
 Fall '10
 Strang
 Linear Algebra, Algebra, Determinant, Factors

Click to edit the document details