Differential Equations and Linear Algebra (4th Edition)
4th Edition
ISBN: 9780321964670
Author: Stephen W. Goode, Scott A. Annin
Publisher: PEARSON
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Chapter 2.5, Problem 5TFR
To determine
Whether the statement “A linear system is consistent if and only if there are free variables in the row-echelon form of the corresponding augmented matrix” is true or false.
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Question 1. Solve the system
-
x1 x2 + 3x3 + 2x4
-x1 + x22x3 + x4
2x12x2+7x3+7x4
Question 2. Consider the system
= 1
=-2
= 1
3x1 - x2 + ax3
= 1
x1 + 3x2 + 2x3
x12x2+2x3
= -b
= 4
1 For what values of a, b will the system be inconsistent?
2 For what values of a, b will the system have only one solution?
For what values of a, b will the saystem have infinitely many solutions?
Chapter 2 Solutions
Differential Equations and Linear Algebra (4th Edition)
Ch. 2.1 - For items a-m, decide if the given statement is...Ch. 2.1 - For items a-m, decide if the given statement is...Ch. 2.1 - For items a-m, decide if the given statement is...Ch. 2.1 - For items a-m, decide if the given statement is...Ch. 2.1 - For items a-m, decide if the given statement is...Ch. 2.1 - For items a-m, decide if the given statement is...Ch. 2.1 - For items a-m, decide if the given statement is...Ch. 2.1 - TRUE-FALSE REVIEW For items (a)(m), decide if the...Ch. 2.1 - TRUE-FALSE REVIEW For items (a)(m), decide if the...Ch. 2.1 - TRUE-FALSE REVIEW For items (a)(m), decide if the...
Ch. 2.1 - TRUE-FALSE REVIEW For items (a)(m), decide if...Ch. 2.1 - TRUE-FALSE REVIEW For items (a)(m), decide if the...Ch. 2.1 - TRUE-FALSE REVIEW For items (a)(m), decide if the...Ch. 2.1 - PROBLEMS If A=[123765023214], determine (a)...Ch. 2.1 - Prob. 2PCh. 2.1 - PROBLEMS For Problems 39, write the matrix with...Ch. 2.1 - PROBLEMS For Problems 39, write the matrix with...Ch. 2.1 - PROBLEMS For Problems 39, write the matrix with...Ch. 2.1 - Prob. 6PCh. 2.1 - PROBLEMS For Problems 39, write the matrix with...Ch. 2.1 - Prob. 8PCh. 2.1 - PROBLEMS For Problems 39, write the matrix with...Ch. 2.1 - For Problems 10-12, determine trA for the given...Ch. 2.1 - For Problems 10-12, determine trA for the given...Ch. 2.1 - For Problems 10-12, determine trA for the given...Ch. 2.1 - For Problems 13-15, write the column vectors and...Ch. 2.1 - For Problems 13-15, write the column vectors and...Ch. 2.1 - For Problems 13-15, write the column vectors and...Ch. 2.1 - If a1=[12],a2=[34],anda3=[51], write the matrix...Ch. 2.1 - If a1=[20411],anda2=[94408], write the matrix...Ch. 2.1 - Prob. 18PCh. 2.1 - PROBLEMS If b1=[214], b2=[576], b3=[000],...Ch. 2.1 - Prob. 20PCh. 2.1 - For Problem 21-26, give an example of a matrix of...Ch. 2.1 - For Problem 21-26, give an example of a matrix of...Ch. 2.1 - For Problem 21-26, give an example of a matrix of...Ch. 2.1 - For Problem 21-26, give an example of a matrix of...Ch. 2.1 - For Problem 21-26, give an example of a matrix of...Ch. 2.1 - Prob. 26PCh. 2.1 - For problems 2730, given an example of a matrix...Ch. 2.1 - For problems 2730, given an example of a matrix...Ch. 2.1 - For problems 2730, given an example of a matrix...Ch. 2.1 - For problems 2730, given an example of a matrix...Ch. 2.1 - Prob. 31PCh. 2.1 - Prob. 32PCh. 2.1 - Show that if A is an nn matrix that is both...Ch. 2.2 - For items a-I, decide if the given statement is...Ch. 2.2 - For items a-I, decide if the given statement is...Ch. 2.2 - Prob. 3TFRCh. 2.2 - Prob. 4TFRCh. 2.2 - For items a-I, decide if the given statement is...Ch. 2.2 - Prob. 6TFRCh. 2.2 - For items a-I, decide if the given statement is...Ch. 2.2 - For items a-I, decide if the given statement is...Ch. 2.2 - For items a-I, decide if the given statement is...Ch. 2.2 - For items a-I, decide if the given statement is...Ch. 2.2 - For items a-I, decide if the given statement is...Ch. 2.2 - For items a-I, decide if the given statement is...Ch. 2.2 - For Problems 1-2, let A=[261103], B=[211044],...Ch. 2.2 - For problems 1-2, let A=[261103], B=[211044],...Ch. 2.2 - Prob. 3PCh. 2.2 - Prob. 4PCh. 2.2 - Let A=[32716035],B=[28108392] and C=[6115]....Ch. 2.2 - Prob. 6PCh. 2.2 - For Problems 6-9, determine Ac by computing an...Ch. 2.2 - Prob. 8PCh. 2.2 - For Problems 6-9, determine Ac by computing an...Ch. 2.2 - Prob. 10PCh. 2.2 - Find A2,A3,andA4 if (a)A=[1123]. (b)A=[010201410].Ch. 2.2 - If A,B,andC are nn matrices, show that a...Ch. 2.2 - If A=[2566], calculate A2 and verify that A...Ch. 2.2 - If A=[104112230], calculate A2 and A3 and verify...Ch. 2.2 - Prob. 15PCh. 2.2 - If A=[x12y], determine all values of x and y for...Ch. 2.2 - The Pauli spin matrices 1,2and3 are defined by...Ch. 2.2 - If A=[1121], B=[3142], find [A,B].Ch. 2.2 - If A1=[1001], A2=[0100], and A3=[0010], compute...Ch. 2.2 - Prob. 20PCh. 2.2 - If A, B and C are nn matrices, find [A,[B,C]] and...Ch. 2.2 - Prob. 22PCh. 2.2 - Prob. 23PCh. 2.2 - Prove property 2 of the identity matrix....Ch. 2.2 - If A and B are nn matrices. Prove that...Ch. 2.2 - Prob. 26PCh. 2.2 - For Problems 26-27, let A=[316], B=[047113],...Ch. 2.2 - Prob. 28PCh. 2.2 - Prob. 29PCh. 2.2 - Prob. 30PCh. 2.2 - Prove that for each positive integer n, there is a...Ch. 2.2 - Use the properties of the transpose to show that B...Ch. 2.2 - Prob. 33PCh. 2.2 - If A is an nn matrix, then the matrices B and C...Ch. 2.2 - Prob. 35PCh. 2.2 - Prob. 36PCh. 2.2 - Prove that if A is an np matrix and...Ch. 2.2 - Prob. 38PCh. 2.2 - If A and B are nn symmetric matrices such that...Ch. 2.2 - Prob. 40PCh. 2.2 - For Problems 41-44, determine the derivative of...Ch. 2.2 - Prob. 42PCh. 2.2 - For Problems 41-44, determine the derivative of...Ch. 2.2 - Prob. 44PCh. 2.2 - Let A=[aij(t)] be an mn matrix function and let...Ch. 2.2 - Prob. 46PCh. 2.2 - Prob. 47PCh. 2.2 - Prob. 48PCh. 2.2 - For Problems 46-49, determine abA(t)dt for the...Ch. 2.2 - Prob. 50PCh. 2.2 - Prob. 51PCh. 2.2 - Prob. 52PCh. 2.2 - Prob. 53PCh. 2.2 - Prob. 54PCh. 2.3 - For items (a)(g), decide if the given statement is...Ch. 2.3 - For items (a)(g), decide if the given statement is...Ch. 2.3 - For items (a)(g), decide if the given statement is...Ch. 2.3 - For items (a)(g), decide if the given statement is...Ch. 2.3 - Prob. 5TFRCh. 2.3 - True-False Review For items a-g, decide if the...Ch. 2.3 - True-False Review For items a-g, decide if the...Ch. 2.3 - For Problems 12, verify that the given triple of...Ch. 2.3 - Prob. 2PCh. 2.3 - Verify that for all values of t, (1t,2+3t,32t) is...Ch. 2.3 - Prob. 4PCh. 2.3 - In Problem 58, make a sketch in the xy-plane in...Ch. 2.3 - Prob. 6PCh. 2.3 - In Problem 58, make a sketch in the xy-plane in...Ch. 2.3 - Prob. 8PCh. 2.3 - For Problems 911, determine the coefficient...Ch. 2.3 - For Problems 911, determine the coefficient...Ch. 2.3 - For Problems 911, determine the coefficient...Ch. 2.3 - Prob. 12PCh. 2.3 - For Problems 12-15, write the system of equations...Ch. 2.3 - Prob. 14PCh. 2.3 - For Problems 12-15, write the system of equations...Ch. 2.3 - For Problems 17-20, write the vector formulation...Ch. 2.3 - Prob. 18PCh. 2.3 - Prob. 19PCh. 2.3 - Prob. 20PCh. 2.3 - Prob. 21PCh. 2.3 - For Problems 21-24, verify that the given vector...Ch. 2.3 - Prob. 23PCh. 2.3 - Prob. 24PCh. 2.4 - True-False Review For items a-i, decide if the...Ch. 2.4 - Prob. 2TFRCh. 2.4 - Prob. 3TFRCh. 2.4 - Prob. 4TFRCh. 2.4 - Prob. 5TFRCh. 2.4 - Prob. 6TFRCh. 2.4 - Prob. 7TFRCh. 2.4 - Prob. 8TFRCh. 2.4 - Prob. 9TFRCh. 2.4 - For Problems 1-8, determine whether the given...Ch. 2.4 - For Problems 1-8, determine whether the given...Ch. 2.4 - For Problems 1-8, determine whether the given...Ch. 2.4 - For Problems 1-8, determine whether the given...Ch. 2.4 - For Problems 1-8, determine whether the given...Ch. 2.4 - For Problems 1-8, determine whether the given...Ch. 2.4 - For Problems 1-8, determine whether the given...Ch. 2.4 - For Problems 1-8, determine whether the given...Ch. 2.4 - For problems 9-18, use elementary row operations...Ch. 2.4 - For problems 9-18, use elementary row operations...Ch. 2.4 - For problems 9-18, use elementary row operations...Ch. 2.4 - For problems 9-18, use elementary row operations...Ch. 2.4 - For problems 9-18, use elementary row operations...Ch. 2.4 - For problems 9-18, use elementary row operations...Ch. 2.4 - For problems 9-18, use elementary row operations...Ch. 2.4 - Problems For Problems 9-18, use elementary row...Ch. 2.4 - Problems For Problems 9-18, use elementary row...Ch. 2.4 - Problems For Problems 9-18, use elementary row...Ch. 2.4 - Problems For Problems 19-26, reduce the given...Ch. 2.4 - Problems For Problems 19-26, reduce the given...Ch. 2.4 - For Problems 19-26, reduce the given matrix to...Ch. 2.4 - For Problems 19-26, reduce the given matrix to...Ch. 2.4 - For Problems 19-26, reduce the given matrix to...Ch. 2.4 - For Problems 19-26, reduce the given matrix to...Ch. 2.4 - For Problems 19-26, reduce the given matrix to...Ch. 2.4 - For Problems 19-26, reduce the given matrix to...Ch. 2.4 - Prob. 27PCh. 2.4 - Prob. 28PCh. 2.4 - Many forms of technology have commands for...Ch. 2.4 - Prob. 30PCh. 2.4 - Prob. 31PCh. 2.4 - Prob. 32PCh. 2.5 - For items a-f, decide if the given statement is...Ch. 2.5 - Prob. 2TFRCh. 2.5 - For items a-f, decide if the given statement is...Ch. 2.5 - Prob. 4TFRCh. 2.5 - Prob. 5TFRCh. 2.5 - For items a-f, decide if the given statement is...Ch. 2.5 - For problems 1-12, use Gaussian elimination to...Ch. 2.5 - For problems 1-12, use Gaussian elimination to...Ch. 2.5 - For problems 1-12, use Gaussian elimination to...Ch. 2.5 - For problems 1-12, use Gaussian elimination to...Ch. 2.5 - For problems 1-12, use Gaussian elimination to...Ch. 2.5 - For problems 1-12, use Gaussian elimination to...Ch. 2.5 - For the Problem 1-12, use Gaussian elimination to...Ch. 2.5 - For the Problem 1-12, use Gaussian elimination to...Ch. 2.5 - For the Problem 1-12, use Gaussian elimination to...Ch. 2.5 - For the Problem 1-12, use Gaussian elimination to...Ch. 2.5 - For Problem 1-12, use Gaussian elimination to...Ch. 2.5 - For Problem 1-12, use Gaussian elimination to...Ch. 2.5 - PROBLEMS For problems 1318, use Gauss-Jordan...Ch. 2.5 - PROBLEMS For problems 1318, use Gauss-Jordan...Ch. 2.5 - PROBLEMS For problems 1318, use Gauss-Jordan...Ch. 2.5 - PROBLEMS For problems 1318, use Gauss-Jordan...Ch. 2.5 - Prob. 17PCh. 2.5 - PROBLEMS For problems 1318, use Gauss-Jordan...Ch. 2.5 - PROBLEMS For problems 1923, determine the solution...Ch. 2.5 - Prob. 20PCh. 2.5 - For Problems 19-23, determine the solution set to...Ch. 2.5 - For Problems 19-23, determine the solution set to...Ch. 2.5 - Prob. 23PCh. 2.5 - PROBLEMS Determine all values of the constant k...Ch. 2.5 - PROBLEMS Determine all values of the constant k...Ch. 2.5 - Prob. 26PCh. 2.5 - Prob. 27PCh. 2.5 - Show that the system...Ch. 2.5 - Prob. 29PCh. 2.5 - Prob. 30PCh. 2.5 - In Problems 3033, use the preceding algorithm to...Ch. 2.5 - Prob. 32PCh. 2.5 - Prob. 33PCh. 2.5 - Prob. 34PCh. 2.5 - Prob. 35PCh. 2.5 - For Problems 36-46, determine the solution set to...Ch. 2.5 - Prob. 37PCh. 2.5 - Prob. 38PCh. 2.5 - Prob. 39PCh. 2.5 - Prob. 40PCh. 2.5 - Prob. 41PCh. 2.5 - Prob. 42PCh. 2.5 - Problems For Problems 3646, determine the solution...Ch. 2.5 - Prob. 44PCh. 2.5 - Problems For Problems 3646, determine the solution...Ch. 2.5 - Prob. 46PCh. 2.5 - Problems For Problems 4757, determine the solution...Ch. 2.5 - Prob. 48PCh. 2.5 - Prob. 49PCh. 2.5 - Prob. 50PCh. 2.5 - Prob. 51PCh. 2.5 - PROBLEMS For Problems 4757, determine the solution...Ch. 2.5 - For Problems 4757, determine the solution set to...Ch. 2.5 - Prob. 54PCh. 2.5 - Prob. 55PCh. 2.5 - For Problem 4757, determine the solution set to...Ch. 2.5 - Prob. 57PCh. 2.6 - TRUE-FALSE REVIEW For items (a)(j), decide if...Ch. 2.6 - TRUE-FALSE REVIEW For items (a)(j), decide if...Ch. 2.6 - Prob. 3TFRCh. 2.6 - TRUE-FALSE REVIEW For items (a)(j), decide if...Ch. 2.6 - Prob. 5TFRCh. 2.6 - TRUE-FALSE REVIEW For items (a)(j), decide if...Ch. 2.6 - TRUE-FALSE REVIEW For items (a)(j), decide if...Ch. 2.6 - Prob. 8TFRCh. 2.6 - Prob. 9TFRCh. 2.6 - TRUE-FALSE REVIEW For items (a)(j), decide if...Ch. 2.6 - For Problems 14, verify by direct multiplication...Ch. 2.6 - For Problems 14, verify by direct multiplication...Ch. 2.6 - Prob. 3PCh. 2.6 - Prob. 4PCh. 2.6 - For Problems 5-18, determine A1, if possible,...Ch. 2.6 - For Problems 5-18, determine A1, if possible,...Ch. 2.6 - PROBLEMS For problems 518, determine A1, if...Ch. 2.6 - Prob. 8PCh. 2.6 - Prob. 9PCh. 2.6 - PROBLEMS For problems 518, determine A1, if...Ch. 2.6 - Prob. 11PCh. 2.6 - PROBLEMS For problems 518, determine A1, if...Ch. 2.6 - PROBLEMS For problems 518, determine A1, if...Ch. 2.6 - PROBLEMS For problems 518, determine A1, if...Ch. 2.6 - PROBLEMS For problems 518, determine A1, if...Ch. 2.6 - PROBLEMS For problems 518, determine A1, if...Ch. 2.6 - Prob. 17PCh. 2.6 - PROBLEMS For problems 518, determine A1, if...Ch. 2.6 - Prob. 19PCh. 2.6 - Prob. 20PCh. 2.6 - PROBLEMS For problems 2126, use A1 to find the...Ch. 2.6 - PROBLEMS For problems 2126, use A1 to find the...Ch. 2.6 - PROBLEMS For problems 2126, use A1 to find the...Ch. 2.6 - PROBLEMS For problems 2126, use A1 to find the...Ch. 2.6 - PROBLEMS For problems 2126, use A1 to find the...Ch. 2.6 - PROBLEMS For problems 2126, use A1 to find the...Ch. 2.6 - PROBLEMS An nn matrix A is called orthogonal if...Ch. 2.6 - Prob. 28PCh. 2.6 - PROBLEMS An nn matrix A is called orthogonal if...Ch. 2.6 - Prob. 30PCh. 2.6 - Prob. 31PCh. 2.6 - PROBLEMS Prove Corollary 2.6.11. Let A1,A2,,Ak be...Ch. 2.6 - For Problems 33-34, use properties of the inverse...Ch. 2.6 - For Problems 33-34, use properties of the inverse...Ch. 2.6 - Prob. 35PCh. 2.6 - Prob. 36PCh. 2.6 - Prob. 37PCh. 2.6 - Prob. 38PCh. 2.6 - Prob. 39PCh. 2.6 - Prob. 40PCh. 2.6 - Prob. 41PCh. 2.6 - Prob. 42PCh. 2.6 - Prob. 43PCh. 2.6 - Let A be an mn matrix with mn, a If rank(A)=m,...Ch. 2.6 - Prob. 45PCh. 2.6 - Prob. 46PCh. 2.6 - Prob. 47PCh. 2.6 - For problems 4749, use built-in functions of some...Ch. 2.6 - Prob. 50PCh. 2.7 - True-False Review: For item a-j decide if the...Ch. 2.7 - Prob. 2TFRCh. 2.7 - True-False Review: For item a-j decide if the...Ch. 2.7 - Prob. 4TFRCh. 2.7 - Prob. 5TFRCh. 2.7 - Prob. 6TFRCh. 2.7 - Prob. 7TFRCh. 2.7 - Prob. 8TFRCh. 2.7 - Prob. 9TFRCh. 2.7 - Prob. 10TFRCh. 2.7 - Problems Write all 33 elementary matrices and...Ch. 2.7 - Prob. 2PCh. 2.7 - Prob. 3PCh. 2.7 - Prob. 4PCh. 2.7 - Prob. 5PCh. 2.7 - Prob. 6PCh. 2.7 - For Problem 7-13, express the matrix A as a...Ch. 2.7 - For Problem 7-13, express the matrix A as a...Ch. 2.7 - Prob. 9PCh. 2.7 - For Problem 7-13, express the matrix A as a...Ch. 2.7 - For Problem 7-13, express the matrix A as a...Ch. 2.7 - Prob. 12PCh. 2.7 - Prob. 13PCh. 2.7 - Prob. 14PCh. 2.7 - Determine a Type 3 lower triangular elementary...Ch. 2.7 - For Problems 16-21, determine the LU factorization...Ch. 2.7 - For Problems 16-21, determine the LU factorization...Ch. 2.7 - For Problems 16-21, determine the LU factorization...Ch. 2.7 - For Problems 16-21, determine the LU factorization...Ch. 2.7 - For Problems 16-21, determine the LU factorization...Ch. 2.7 - For Problem 16-21 determine the LU factorization...Ch. 2.7 - For Problems 22-25, use the LU factorization of A...Ch. 2.7 - For Problems 22-25, use the LU factorization of A...Ch. 2.7 - For Problems 22-25, use the LU factorization of A...Ch. 2.7 - For Problems 22-25, use the LU factorization of A...Ch. 2.7 - Use the LU factorization of A=[1223] to solve each...Ch. 2.7 - Use the LU factorization of A=[142314571] to solve...Ch. 2.7 - If P=P1P2Pk where each Pi is an elementary...Ch. 2.7 - Prove that a. the inverse of an invertible upper...Ch. 2.7 - In this problem, we prove that the LU...Ch. 2.7 - Prob. 31PCh. 2.8 - True-False Review: For items a-d, decide if the...Ch. 2.8 - Prob. 2TFRCh. 2.8 - Prob. 3TFRCh. 2.8 - Prob. 4TFRCh. 2.8 - Problems: Use part c of the Invertible Matrix...Ch. 2.8 - Problems: Give a direct proof of the fact that...Ch. 2.8 - Problems: Give a direct proof of the fact that...Ch. 2.8 - Problems: Use the equivalence of a and e in the...Ch. 2.9 - Let A=[24261150], B=[30221301], C=[5631]. For...Ch. 2.9 - Let A=[24261150], B=[30221301], C=[5631]. For...Ch. 2.9 - Let A=[24261150], B=[30221301], C=[5631]. For...Ch. 2.9 - Prob. 4APCh. 2.9 - Prob. 5APCh. 2.9 - Prob. 6APCh. 2.9 - Prob. 7APCh. 2.9 - Prob. 8APCh. 2.9 - Prob. 9APCh. 2.9 - Prob. 10APCh. 2.9 - Prob. 11APCh. 2.9 - Prob. 12APCh. 2.9 - Prob. 13APCh. 2.9 - Prob. 14APCh. 2.9 - Prob. 15APCh. 2.9 - For Problem 16-19 let...Ch. 2.9 - Prob. 17APCh. 2.9 - Prob. 18APCh. 2.9 - For Problem 16-19 let...Ch. 2.9 - For Problem 20-26 determine the solutions set the...Ch. 2.9 - For Problem 20-26 determine the solutions set the...Ch. 2.9 - For problems 20-26, determine the solution set to...Ch. 2.9 - Prob. 23APCh. 2.9 - Prob. 24APCh. 2.9 - Prob. 25APCh. 2.9 - Prob. 26APCh. 2.9 - For problems 27-30, determine all values of k for...Ch. 2.9 - For problems 27-30, determine all values of k for...Ch. 2.9 - Prob. 29APCh. 2.9 - For problems 27-30, determine all values of k for...Ch. 2.9 - Prob. 31APCh. 2.9 - For problems 32-37, a find a row-echelon form of...Ch. 2.9 - For problems 32-37, a find a row-echelon form of...Ch. 2.9 - For problems 32-37, a find a row-echelon form of...Ch. 2.9 - Prob. 35APCh. 2.9 - Prob. 36APCh. 2.9 - Prob. 37APCh. 2.9 - Prob. 38APCh. 2.9 - Prob. 39APCh. 2.9 - Prob. 40APCh. 2.9 - Prob. 41APCh. 2.9 - Prob. 42APCh. 2.9 - For problems 42-45, a express the given matrix as...Ch. 2.9 - For problems 42-45, a express the given matrix as...Ch. 2.9 - For problems 42-45, a express the given matrix as...Ch. 2.9 - Prob. 46APCh. 2.9 - Prob. 47APCh. 2.9 - Prob. 48APCh. 2.9 - Prob. 49APCh. 2.9 - Prob. 50APCh. 2.9 - Prob. 51AP
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