Student Solutions Manual for Gallian's Contemporary Abstract Algebra, 9th
9th Edition
ISBN: 9781305657977
Author: Gallian, Joseph
Publisher: Brooks Cole
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Chapter 0, Problem 11E
Let n and a be positive integers and let
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ANALYZING RELATIONSHIPS Describe the x-values for which (a) f is increasing or decreasing, (b) f(x) > 0 and (c) f(x) <0.
y Af
-2
1
2 4x
a. The function is increasing when
and
decreasing when
By forming the augmented matrix corresponding to this system of equations and usingGaussian elimination, find the values of t and u that imply the system:(i) is inconsistent.(ii) has infinitely many solutions.(iii) has a unique solutiona=2 b=1
if a=2 and b=1
1) Calculate 49(B-1)2+7B−1AT+7ATB−1+(AT)2
2)Find a matrix C such that (B − 2C)-1=A
3) Find a non-diagonal matrix E ̸= B such that det(AB) = det(AE)
Chapter 0 Solutions
Student Solutions Manual for Gallian's Contemporary Abstract Algebra, 9th
Ch. 0 - For n=5 , 8, 12, 20, and 25, find all positive...Ch. 0 - Determine a. gcd(2,10) lcm(2,10) b. gcd(20,8)...Ch. 0 - Determine 51 mod 13, 342 mod 85, 62 mod 15, 10 mod...Ch. 0 - Find integers s and t such that 1=7s+11t ? t. Show...Ch. 0 - Show that if a and b are positive integers, then...Ch. 0 - Suppose a and b are integers that divide the...Ch. 0 - If a and b are integers and n is a positive...Ch. 0 - Let d=gcd(a,b) . If a=da and b=db , show that...Ch. 0 - Let n be a fixed positive integer greater than 1....Ch. 0 - Let a and b be positive integers and let...
Ch. 0 - Let n and a be positive integers and let...Ch. 0 - Show that 5n+3and7n+4 are relatively prime for all...Ch. 0 - Suppose that m and n are relatively prime and r is...Ch. 0 - Let p, q, and r be primes other than 3. Show that...Ch. 0 - Prove that every prime greater than 3 can be...Ch. 0 - Determine 71000 mod 6 and 61001 mod 7.Ch. 0 - Let a, b, s, and t be integers. If a mod...Ch. 0 - Determine 8402 mod 5.Ch. 0 - Show that gcd(a,bc)=1 if and only if gcd(a,b)=1...Ch. 0 - Let p1,p2,...,pn be primes. Show that p1p2pn+1 is...Ch. 0 - Prove that there are infinitely many primes....Ch. 0 - Prob. 22ECh. 0 - Prob. 23ECh. 0 - For any complex numbers z1andz2 prove that...Ch. 0 - Give an “if and only if” statement that describes...Ch. 0 - For inputs of 0 and 1 and mod 2 arithmetic...Ch. 0 - For every positive integer n, prove that a set...Ch. 0 - Prove that 2n32n1 is always divisible by 17.Ch. 0 - Prove that there is some positive integer n such...Ch. 0 - (Generalized Euclid’s Lemma) If p is a prime and p...Ch. 0 - Prob. 31ECh. 0 - What is the largest bet that cannot be made with...Ch. 0 - Prove that the First Principle of Mathematical...Ch. 0 - The Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13,...Ch. 0 - Prove by induction on n that for all positive...Ch. 0 - Suppose that there is a statement involving a...Ch. 0 - In the cut “As” from Songs in the Key of Life,...Ch. 0 - Prove that for every integer n, n3 mod 6=n mod 6.Ch. 0 - If it is 2:00 A.M. now, what time will it be 3736...Ch. 0 - Determine the check digit for a money order with...Ch. 0 - Suppose that in one of the noncheck positions of a...Ch. 0 - Suppose that a money order identification number...Ch. 0 - A transposition error involving distinct adjacent...Ch. 0 - Determine the check digit for the Avis rental car...Ch. 0 - Show that a substitution of a digit ai for the...Ch. 0 - Determine which transposition errors involving...Ch. 0 - Use the UPC scheme to determine the check digit...Ch. 0 - Explain why the check digit for a money order for...Ch. 0 - The 10-digit International Standard Book Number...Ch. 0 - Suppose that an ISBN-10 has a smudged entry where...Ch. 0 - Suppose three consecutive digits abc of an ISBN-10...Ch. 0 - Prob. 52ECh. 0 - Suppose the weighting vector for ISBN-10s were...Ch. 0 - Use the two-check-digit error-correction method...Ch. 0 - Suppose that an eight-digit number has two check...Ch. 0 - The state of Utah appends a ninth digit a9 to an...Ch. 0 - Complete the proof of Theorem 0.8.Ch. 0 - Let S be the set of real numbers. If a,bS , define...Ch. 0 - Let S be the set of integers. If a,bS , define aRb...Ch. 0 - Let S be the set of integers. If a,bS , define aRb...Ch. 0 - Complete the proof of Theorem 0.7 by showing that...Ch. 0 - Prove that 3, 5, and 7 are the only three...Ch. 0 - What is the last digit of 3100 ? What is the last...Ch. 0 - Prove that there are no rational numbers x and y...Ch. 0 - (Cancellation Property) Suppose , and are...
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- Write the equation line shown on the graph in slope, intercept form.arrow_forward1.2.15. (!) Let W be a closed walk of length at least 1 that does not contain a cycle. Prove that some edge of W repeats immediately (once in each direction).arrow_forward1.2.18. (!) Let G be the graph whose vertex set is the set of k-tuples with elements in (0, 1), with x adjacent to y if x and y differ in exactly two positions. Determine the number of components of G.arrow_forward
- 1.2.17. (!) Let G,, be the graph whose vertices are the permutations of (1,..., n}, with two permutations a₁, ..., a,, and b₁, ..., b, adjacent if they differ by interchanging a pair of adjacent entries (G3 shown below). Prove that G,, is connected. 132 123 213 312 321 231arrow_forward1.2.19. Let and s be natural numbers. Let G be the simple graph with vertex set Vo... V„−1 such that v; ↔ v; if and only if |ji| Є (r,s). Prove that S has exactly k components, where k is the greatest common divisor of {n, r,s}.arrow_forward1.2.20. (!) Let u be a cut-vertex of a simple graph G. Prove that G - v is connected. עarrow_forward
- 1.2.12. (-) Convert the proof at 1.2.32 to an procedure for finding an Eulerian circuit in a connected even graph.arrow_forward1.2.16. Let e be an edge appearing an odd number of times in a closed walk W. Prove that W contains the edges of a cycle through c.arrow_forward1.2.11. (−) Prove or disprove: If G is an Eulerian graph with edges e, f that share vertex, then G has an Eulerian circuit in which e, f appear consecutively. aarrow_forward
- By forming the augmented matrix corresponding to this system of equations and usingGaussian elimination, find the values of t and u that imply the system:(i) is inconsistent.(ii) has infinitely many solutions.(iii) has a unique solutiona=2 b=1arrow_forward1.2.6. (-) In the graph below (the paw), find all the maximal paths, maximal cliques, and maximal independent sets. Also find all the maximum paths, maximum cliques, and maximum independent sets.arrow_forward1.2.13. Alternative proofs that every u, v-walk contains a u, v-path (Lemma 1.2.5). a) (ordinary induction) Given that every walk of length 1-1 contains a path from its first vertex to its last, prove that every walk of length / also satisfies this. b) (extremality) Given a u, v-walk W, consider a shortest u, u-walk contained in W.arrow_forward
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