Solved problems on matrices, rearrangement, generating functions, equations, machines and Turing machines
7 Pages1484 Words309 Views
Added on 2019-09-22
About This Document
This article provides solutions to problems on matrices, rearrangement, generating functions, equations, machines and Turing machines. It includes a closed form for the generating function for each sequence, a Turing Machine to construct the function f(n) = 3 [1/3n] + 2 and solutions to problems on matrices, rearrangement, generating functions, equations, machines and Turing machines.
BookmarkShareRelated Documents
Assignment1. For each of the following matrices find the multiplicative inverse, or explain why it doesn’t have one.(a)[3523]Solution: A= [3523]|A| = 3*3 – 2*5 = 9-10= -1Minors areM11 = 3, M12=2, M21=5, M22=3Co factors of matrix areC11 = C11C12 = -C12C21 = -C21C22 = C22Cofactors Matrix [3−2−53]A-1 = 1/|A| * (Cofactor)TA-1 = 1/-1 * [3−5−23]A-1 = [−352−3](b)[236324669]Solution: A= [236324669]|A| = 2*(2*9-6*4)– 3*(3*9-6*4) +6*(3*6-6*2) = 2*(-6)- 3*(3) +6*(6) = 15Minors areM11=-6, M12=3, M13=6,M21=-9, M22=-18, M23=-6,M31=0, M32=-10, M33=-2
A-1 = 1/1 * [1−21101−36001−40001]A-1 = [1−21101−36001−40001]2. How many ways can we rearrange the letters a b c d e f g h i j so that no vowel endsup in the position where it began?Solution: n=10 (a, b, c, d, e, f, g, h, i, j) Vowels a, e, i (3 vowels)No. of Ways= 2419143. Find a closed form for the generating function for each of these sequences.(a) 7, 3, 4, 6, 7, 3, 4, 6, 7, 3, 4, 6, . . .Solution: 1 mod 4 = 12 mod 4 = 23 mod 4 = 34 mod 4 = 05 mod 4 = 16 mod 4 = 27 mod 4 = 38 mod 4 = 0 And so on...F (n) = {ifnmod4=1,7ifnmod4=2,3ifnmod4=3,3ifnmod4=0,6} n E |N(b) .1, 0.01, 0.001, 0.0001, . . . Solution: a2/a1= a3/a2=...... an/an-1= 1/10a1= 1/10f(n) = an/an-1= 1/10; n≥2; n E |N; a1=1/10(c) 2, 5, 8, 11, 14, 17, 20, . . .