Linear Algebra and Its Applications (5th Edition)
5th Edition
ISBN: 9780321982384
Author: David C. Lay, Steven R. Lay, Judi J. McDonald
Publisher: PEARSON
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Chapter 3.3, Problem 18E
Suppose that all the entries in A are integers and det A = 1. Explain why all the entries in A−1 are integers.
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39. The balls shown have different masses and speeds. Rank
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12.0 m/s
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X
Simplify the below expression.
3 - (-7)
(6) ≤
a) Determine the following groups:
Homz(Q, Z),
Homz(Q, Q),
Homz(Q/Z, Z)
for n E N.
Homz(Z/nZ, Q)
b) Show for ME MR: HomR (R, M) = M.
Chapter 3 Solutions
Linear Algebra and Its Applications (5th Edition)
Ch. 3.1 - Compute |5722030458030506|.Ch. 3.1 - Compute the determinants in Exercises 18 using a...Ch. 3.1 - Compute the determinants in Exercises 18 using a...Ch. 3.1 - Compute the determinants in Exercises 18 using a...Ch. 3.1 - Compute the determinants in Exercises 18 using a...Ch. 3.1 - Compute the determinants in Exercises 18 using a...Ch. 3.1 - Compute the determinants in Exercises 18 using a...Ch. 3.1 - Compute the determinants in Exercises 18 using a...Ch. 3.1 - Compute the determinants in Exercises 18 using a...Ch. 3.1 - Compute the determinants in Exercises 914 by a...
Ch. 3.1 - Compute the determinants in Exercises 914 by a...Ch. 3.1 - Compute the determinants in Exercises 914 by a...Ch. 3.1 - Compute the determinants in Exercises 914 by...Ch. 3.1 - Compute the determinants in Exercises 914 by...Ch. 3.1 - Compute the determinants in Exercises 914 by...Ch. 3.1 - The expansion of a 3 3 determinant can be...Ch. 3.1 - The expansion of a 3 3 determinant can be...Ch. 3.1 - The expansion of a 3 3 determinant can be...Ch. 3.1 - The expansion of a 3 3 determinant can be...Ch. 3.1 - In Exercises 1924, explore the effect of an...Ch. 3.1 - In Exercises 1924, explore the effect of an...Ch. 3.1 - In Exercises 1924, explore the effect of an...Ch. 3.1 - In Exercises 1924, explore the effect of an...Ch. 3.1 - In Exercises 1924, explore the effect of an...Ch. 3.1 - In Exercises 1924, explore the effect of an...Ch. 3.1 - Compute the determinants of the elementary...Ch. 3.1 - Compute the determinants of the elementary...Ch. 3.1 - Compute the determinants of the elementary...Ch. 3.1 - Compute the determinants of the elementary...Ch. 3.1 - Compute the determinants of the elementary...Ch. 3.1 - Compute the determinants of the elementary...Ch. 3.1 - Use Exercises 2528 to answer the questions in...Ch. 3.1 - Use Exercises 2528 to answer the questions in...Ch. 3.1 - In Exercises 3336, verify that det EA = (det...Ch. 3.1 - In Exercises 3336, verify that det EA = (det...Ch. 3.1 - In Exercises 3336, verify that det EA = (det...Ch. 3.1 - In Exercises 3336, verify that det EA = (det...Ch. 3.1 - Let A = [3142] Write 5A. Is det 5A = 5 det A?Ch. 3.1 - Let .A = [abcd] and let k be a scalar. Find a...Ch. 3.1 - In Exercises 39 and 40, A is an n n matrix. Mark...Ch. 3.1 - a. The cofactor expansion of det A down a column...Ch. 3.1 - Let u = [30] and v = [12]. Compute the area of the...Ch. 3.1 - Let u = [ab] and v = [c0], where a, b, and c are...Ch. 3.2 - PRACTICE PROBLEMS 1. Compute |13122512045131068|...Ch. 3.2 - Use a determinant to decide if v1, v2, and v3 are...Ch. 3.2 - Let A be an n n matrix such that A2 = I. Show...Ch. 3.2 - Each equation in Exercises 14 illustrates a...Ch. 3.2 - Each equation in Exercises 14 illustrates a...Ch. 3.2 - Each equation in Exercises 14 illustrates a...Ch. 3.2 - Each equation in Exercises 14 illustrates a...Ch. 3.2 - Find the determinants in Exercises 510 by row...Ch. 3.2 - Find the determinants in Exercises 510 by row...Ch. 3.2 - Find the determinants in Exercises 510 by row...Ch. 3.2 - Find the determinants in Exercises 510 by row...Ch. 3.2 - Find the determinants in Exercises 510 by row...Ch. 3.2 - Find the determinants in Exercises 510 by row...Ch. 3.2 - Combine the methods of row reduction and cofactor...Ch. 3.2 - Combine the methods of row reduction and cofactor...Ch. 3.2 - Combine the methods of row reduction and cofactor...Ch. 3.2 - Combine the methods of row reduction and cofactor...Ch. 3.2 - Find the determinants in Exercises 1520, where 15....Ch. 3.2 - Find the determinants in Exercises 1520, where 16....Ch. 3.2 - Find the determinants in Exercises 1520, where...Ch. 3.2 - Find the determinants in Exercises 1520, where...Ch. 3.2 - Find the determinants in Exercises 1520, where...Ch. 3.2 - Find the determinants in Exercises 1520, where...Ch. 3.2 - In Exercises 2123, use determinants to find out if...Ch. 3.2 - In Exercises 2123, use determinants to find out if...Ch. 3.2 - In Exercises 2123, use determinants to find out if...Ch. 3.2 - In Exercises 2426, use determinants to decide if...Ch. 3.2 - In Exercises 2426, use determinants to decide if...Ch. 3.2 - In Exercises 2426, use determinants to decide if...Ch. 3.2 - In Exercises 27 and 28, A and B are n n matrices....Ch. 3.2 - a. If three row interchanges are made in...Ch. 3.2 - Compute det B4 where B = [101112121]Ch. 3.2 - Use Theorem 3 (but not Theorem 4) to show that if...Ch. 3.2 - Show that if A is invertible, then detA1=1detA.Ch. 3.2 - Suppose that A is a square matrix such that det A3...Ch. 3.2 - Let A and B be square matrices. Show that even...Ch. 3.2 - Let A and P be square matrices, with P invertible....Ch. 3.2 - Let U be a square matrix such that UTU = 1. Show...Ch. 3.2 - Find a formula for det(rA) when A is an n n...Ch. 3.2 - Verify that det AB = (det A)(det B) for the...Ch. 3.2 - Verify that det AB = (det A)(det B) for the...Ch. 3.2 - Let A and B be 3 3 matrices, with det A = 3 and...Ch. 3.2 - Let A and B be 4 4 matrices, with det A = 3 and...Ch. 3.2 - Prob. 41ECh. 3.2 - Let A = [1001] and B = [abcd]. Show that det(A +...Ch. 3.2 - Verify that det A = det B + det C, where A =...Ch. 3.2 - Right-multiplication by an elementary matrix E...Ch. 3.3 - Let S be the parallelogram determined by the...Ch. 3.3 - Use Cramers rule to compute the solutions of the...Ch. 3.3 - Use Cramers rule to compute the solutions of the...Ch. 3.3 - Use Cramers rule to compute the solutions of the...Ch. 3.3 - Use Cramers rule to compute the solutions of the...Ch. 3.3 - Use Cramers rule to compute the solutions of the...Ch. 3.3 - Use Cramers rule to compute the solutions of the...Ch. 3.3 - In Exercises 710, determine the values of the...Ch. 3.3 - In Exercises 710, determine the values of the...Ch. 3.3 - In Exercises 710, determine the values of the...Ch. 3.3 - In Exercises 710, determine the values of the...Ch. 3.3 - In Exercises 1116, compute the adjugate of the...Ch. 3.3 - In Exercises 1116, compute the adjugate of the...Ch. 3.3 - In Exercises 1116, compute the adjugate of the...Ch. 3.3 - In Exercises 1116, compute the adjugate of the...Ch. 3.3 - In Exercises 1116, compute the adjugate of the...Ch. 3.3 - In Exercises 1116, compute the adjugate of the...Ch. 3.3 - Show that if A is 2 2, then Theorem 8 gives the...Ch. 3.3 - Suppose that all the entries in A are integers and...Ch. 3.3 - In Exercises 1922, find the area of the...Ch. 3.3 - In Exercises 1922, find the area of the...Ch. 3.3 - In Exercises 1922, find the area of the...Ch. 3.3 - In Exercises 19-22, find the area of the...Ch. 3.3 - Find the volume of the parallelepiped with one...Ch. 3.3 - Find the volume of the parallelepiped with one...Ch. 3.3 - Use the concept of volume to explain why the...Ch. 3.3 - Let T : m n be a linear transformation, and let p...Ch. 3.3 - Let S be the parallelogram determined by the...Ch. 3.3 - Repeat Exercise 27 with b1=[47], b2=[01], and...Ch. 3.3 - Find a formula for the area of the triangle whose...Ch. 3.3 - Let R be the triangle with vertices at (x1, y1),...Ch. 3.3 - Let T: 3 3 be the linear transformation...Ch. 3.3 - Let S be the tetrahedron in 3 with vertices at the...Ch. 3 - Mark each statement True or False. Justify each...Ch. 3 - Use row operations to show that the determinants...Ch. 3 - Use row operations to show that the determinants...Ch. 3 - Prob. 4SECh. 3 - Compute the determinants in Exercises 5 and 6. 5....Ch. 3 - Compute the determinants in Exercises 5 and 6. 6....Ch. 3 - Show that the equation of the line in 2 through...Ch. 3 - Prob. 8SECh. 3 - Exercise 9 and 10 concern determinants of the...Ch. 3 - Let f(t) = det V, with x1, x2, and x3 all...Ch. 3 - Find the area of the parallelogram determined by...Ch. 3 - Use the concept of area of a parallelogram to...Ch. 3 - Prob. 13SECh. 3 - Let A,B,C,D, and I be n n matrices. Use the...Ch. 3 - Let A, B, C, and D be n n matrices with A...Ch. 3 - Let J be the n n matrix of all 1s, and consider A...Ch. 3 - Prob. 17SE
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- 1. If f(x² + 1) = x + 5x² + 3, what is f(x² - 1)?arrow_forward2. What is the total length of the shortest path that goes from (0,4) to a point on the x-axis, then to a point on the line y = 6, then to (18.4)?arrow_forwardموضوع الدرس Prove that Determine the following groups Homz(QZ) Hom = (Q13,Z) Homz(Q), Hom/z/nZ, Qt for neN- (2) Every factor group of adivisible group is divisble. • If R is a Skew ficald (aring with identity and each non Zero element is invertible then every R-module is free.arrow_forward
- Please help me with these questions. I am having a hard time understanding what to do. Thank youarrow_forwardAnswersarrow_forward************* ********************************* Q.1) Classify the following statements as a true or false statements: a. If M is a module, then every proper submodule of M is contained in a maximal submodule of M. b. The sum of a finite family of small submodules of a module M is small in M. c. Zz is directly indecomposable. d. An epimorphism a: M→ N is called solit iff Ker(a) is a direct summand in M. e. The Z-module has two composition series. Z 6Z f. Zz does not have a composition series. g. Any finitely generated module is a free module. h. If O→A MW→ 0 is short exact sequence then f is epimorphism. i. If f is a homomorphism then f-1 is also a homomorphism. Maximal C≤A if and only if is simple. Sup Q.4) Give an example and explain your claim in each case: Monomorphism not split. b) A finite free module. c) Semisimple module. d) A small submodule A of a module N and a homomorphism op: MN, but (A) is not small in M.arrow_forward
- I need diagram with solutionsarrow_forwardT. Determine the least common denominator and the domain for the 2x-3 10 problem: + x²+6x+8 x²+x-12 3 2x 2. Add: + Simplify and 5x+10 x²-2x-8 state the domain. 7 3. Add/Subtract: x+2 1 + x+6 2x+2 4 Simplify and state the domain. x+1 4 4. Subtract: - Simplify 3x-3 x²-3x+2 and state the domain. 1 15 3x-5 5. Add/Subtract: + 2 2x-14 x²-7x Simplify and state the domain.arrow_forwardQ.1) Classify the following statements as a true or false statements: Q a. A simple ring R is simple as a right R-module. b. Every ideal of ZZ is small ideal. very den to is lovaginz c. A nontrivial direct summand of a module cannot be large or small submodule. d. The sum of a finite family of small submodules of a module M is small in M. e. The direct product of a finite family of projective modules is projective f. The sum of a finite family of large submodules of a module M is large in M. g. Zz contains no minimal submodules. h. Qz has no minimal and no maximal submodules. i. Every divisible Z-module is injective. j. Every projective module is a free module. a homomorp cements Q.4) Give an example and explain your claim in each case: a) A module M which has a largest proper submodule, is directly indecomposable. b) A free subset of a module. c) A finite free module. d) A module contains no a direct summand. e) A short split exact sequence of modules.arrow_forward
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