Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)
5th Edition
ISBN: 9780134689531
Author: Lee Johnson, Dean Riess, Jimmy Arnold
Publisher: PEARSON
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Textbook Question
Chapter 4.4, Problem 8E
In Exercises 1 – 14, find the characteristic polynomial and the eigenvalues for the given matrix. Also, give the algebraic multiplicity of each eigenvalue. [Note: In each case the eigenvalues are integers.]
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a
Question 7. If det d e f
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4
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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?
Question 5. Let A, B, C ben x n-matrices, S is nonsigular. If A = S-1 BS, show that
det (A) = det (B)
Question 6. For what values of k is the matrix A = (2- k
-1
-1
2) singular?
k
Chapter 4 Solutions
Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)
Ch. 4.1 - In Exercises 1-12, find the eigenvalues and the...Ch. 4.1 - In Exercises 1-12, find the eigenvalues and the...Ch. 4.1 - In Exercises 1-12, find the eigenvalues and the...Ch. 4.1 - In Exercises 1-12, find the eigenvalues and the...Ch. 4.1 - In Exercises 1-12, find the eigenvalues and the...Ch. 4.1 - In Exercises 1-12, find the eigenvalues and the...Ch. 4.1 - In Exercises 1-12, find the eigenvalues and the...Ch. 4.1 - In Exercises 1-12, find the eigenvalues and the...Ch. 4.1 - In Exercises 1-12, find the eigenvalues and the...Ch. 4.1 - In Exercises 1-12, find the eigenvalues and the...
Ch. 4.1 - In Exercises 1-12, find the eigenvalues and the...Ch. 4.1 - In Exercises 1-12, find the eigenvalues and the...Ch. 4.1 - Using Eq.4, apply the singularity test to the...Ch. 4.1 - Using Eq.4, apply the singularity test to the...Ch. 4.1 - Using Eq.4, apply the singularity test to the...Ch. 4.1 - Using Eq.4, apply the singularity test to the...Ch. 4.1 - Consider the (22) symmetric matrix A=[abbd]. Show...Ch. 4.1 - Consider the (22) matrix A given by A=[abba],b0....Ch. 4.1 - Let A be a (22) matrix. Show that A and AT have...Ch. 4.2 - In Exercises 1-6, list the minor matrix Mij, and...Ch. 4.2 - In Exercises 1-6, list the minor matrix Mij, and...Ch. 4.2 - Prob. 3ECh. 4.2 - In Exercises 1-6, list the minor matrix Mij, and...Ch. 4.2 - Prob. 5ECh. 4.2 - In Exercises 1-6, list the minor matrix Mij, and...Ch. 4.2 - Prob. 7ECh. 4.2 - Prob. 8ECh. 4.2 - Prob. 9ECh. 4.2 - Prob. 10ECh. 4.2 - Prob. 11ECh. 4.2 - In Exercises 8-19, calculate the determinant of...Ch. 4.2 - Prob. 13ECh. 4.2 - In Exercises 8-19, calculate the determinant of...Ch. 4.2 - In Exercises 8-19, calculate the determinant of...Ch. 4.2 - In Exercises 8-19, calculate the determinant of...Ch. 4.2 - Prob. 17ECh. 4.2 - In Exercises 8-19, calculate the determinant of...Ch. 4.2 - Prob. 19ECh. 4.2 - Let A=(aij) be a given (33) matrix. Form the...Ch. 4.2 - In Exercises 21 and 22, find all ordered pairs...Ch. 4.2 - In Exercises 21 and 22, find all ordered pairs...Ch. 4.2 - Let A=(aij) be the (nn) matrix specified thus:...Ch. 4.2 - Let A and B be (nn) matrices. Use Theorems 2 and 3...Ch. 4.2 - Suppose that A is a (nn) nonsingular matrix, and...Ch. 4.2 - Prob. 26ECh. 4.2 - In Exercises 27-30, use Theorem 2 and Exercise 25...Ch. 4.2 - In Exercises 27-30, use Theorem 2 and Exercise 25...Ch. 4.2 - In Exercises 27-30, use Theorem 2 and Exercise 25...Ch. 4.2 - In Exercises 27-30, use Theorem 2 and Exercise 25...Ch. 4.2 - a Let A be an (nn) matrix. If n=3, det(A) can be...Ch. 4.2 - Prob. 32ECh. 4.2 - Prob. 33ECh. 4.2 - Prob. 34ECh. 4.3 - In Exercise 1-6, evaluate det(A) by using row...Ch. 4.3 - In Exercise 1-6, evaluate det(A) by using row...Ch. 4.3 - Prob. 3ECh. 4.3 - In Exercise 1-6, evaluate det(A) by using row...Ch. 4.3 - Prob. 5ECh. 4.3 - Prob. 6ECh. 4.3 - Prob. 7ECh. 4.3 - In Exercise 7-12, use only column interchanges or...Ch. 4.3 - Prob. 9ECh. 4.3 - In Exercise 7-12, use only column interchanges or...Ch. 4.3 - In Exercise 7-12, use only column interchanges or...Ch. 4.3 - Prob. 12ECh. 4.3 - In Exercise 13-18, assume that the (33) matrix A...Ch. 4.3 - In Exercise 13-18, assume that the (33) matrix A...Ch. 4.3 - In Exercise 13-18, assume that the (33) matrix A...Ch. 4.3 - In Exercise 13-18, assume that the (33) matrix A...Ch. 4.3 - Prob. 17ECh. 4.3 - Prob. 18ECh. 4.3 - In Exercise 19-22, evaluate the (44) determinants....Ch. 4.3 - In Exercise 19-22, evaluate the (44) determinants....Ch. 4.3 - In Exercise 19-22, evaluate the (44) determinants....Ch. 4.3 - In Exercise 19-22, evaluate the (44) determinants....Ch. 4.3 - In Exercise 23 and 24, use row operations to...Ch. 4.3 - In Exercise 23 and 24, use row operations to...Ch. 4.3 - Let A be a (nn) matrix. Use Theorem 7 to argue...Ch. 4.3 - Prove the corollary to Theorem 6. Hint: Suppose...Ch. 4.3 - Find examples of (22) matrices A and B such that...Ch. 4.3 - An (nn) matrix A is called skew symmetric if AT=A....Ch. 4.4 - In Exercises 1 14, find the characteristic...Ch. 4.4 - In Exercises 1 14, find the characteristic...Ch. 4.4 - In Exercises 1 14, find the characteristic...Ch. 4.4 - In Exercises 1 14, find the characteristic...Ch. 4.4 - In Exercises 1 14, find the characteristic...Ch. 4.4 - In Exercises 1 14, find the characteristic...Ch. 4.4 - In Exercises 1 14, find the characteristic...Ch. 4.4 - In Exercises 1 14, find the characteristic...Ch. 4.4 - In Exercises 1 14, find the characteristic...Ch. 4.4 - In Exercises 1 14, find the characteristic...Ch. 4.4 - In Exercises 1 14, find the characteristic...Ch. 4.4 - In Exercises 1 14, find the characteristic...Ch. 4.4 - In Exercises 1 14, find the characteristic...Ch. 4.4 - In Exercises 1 14, find the characteristic...Ch. 4.4 - Prove property b of theorem 11. Hint: Begin with...Ch. 4.4 - Prove property c of Theorem 11. Theorem 11 Let A...Ch. 4.4 - Complete the proof of property a of Theorem 11....Ch. 4.4 - Let qt=t3-2t2-t+2; and for any nn matrix H, define...Ch. 4.4 - With qt as in Exercise 18, verify that qC is the...Ch. 4.4 - Exercises 20 23 illustrate the Cayley-Hamilton...Ch. 4.4 - Exercises 20 23 illustrate the Cayley-Hamilton...Ch. 4.4 - Exercises 20 23 illustrate the Cayley-Hamilton...Ch. 4.4 - Exercises 20 23 illustrate the Cayley-Hamilton...Ch. 4.4 - This problem establishes a special case of the...Ch. 4.4 - Consider the 22 matrix A given by A=abcd. The...Ch. 4.4 - Prob. 26ECh. 4.4 - Let qt=tn+an-1tn-1++a1t+a0, and define the nn...Ch. 4.4 - Prob. 28ECh. 4.4 - Prob. 29ECh. 4.4 - Prob. 30ECh. 4.4 - Prob. 31ECh. 4.4 - Prob. 32ECh. 4.4 - Prob. 33ECh. 4.4 - Prob. 34ECh. 4.5 - The following list of matrices and their...Ch. 4.5 - The following list of matrices and their...Ch. 4.5 - The following list of matrices and their...Ch. 4.5 - The following list of matrices and their...Ch. 4.5 - The following list of matrices and their...Ch. 4.5 - The following list of matrices and their...Ch. 4.5 - The following list of matrices and their...Ch. 4.5 - The following list of matrices and their...Ch. 4.5 - The following list of matrices and their...Ch. 4.5 - The following list of matrices and their...Ch. 4.5 - The following list of matrices and their...Ch. 4.5 - In Exercise 12-17, find the eigenvalues and the...Ch. 4.5 - In Exercise 12-17, find the eigenvalues and the...Ch. 4.5 - In Exercise 12-17, find the eigenvalues and the...Ch. 4.5 - In Exercise 12-17, find the eigenvalues and the...Ch. 4.5 - In Exercise 12-17, find the eigenvalues and the...Ch. 4.5 - In Exercise 12-17, find the eigenvalues and the...Ch. 4.5 - If a vector x is a linear combination of...Ch. 4.5 - As in Exercise 18, calculate A10x for...Ch. 4.5 - Consider a (44) matrix H of the form...Ch. 4.5 - An (nn) matrix P is called idempotent if P2=P....Ch. 4.5 - Let P be an idempotent matrix. Show that the only...Ch. 4.5 - Let u be a vector in Rn such that uTu=1. Show that...Ch. 4.5 - Verify that if Q is idempotent, then so is IQ....Ch. 4.5 - Suppose that u and v are vectors in Rn such that...Ch. 4.5 - Show that any nonzero vector of the form au+bv is...Ch. 4.5 - Prob. 27ECh. 4.5 - Let A be a symmetric matrix and suppose that Au=u,...Ch. 4.5 - Prob. 29ECh. 4.6 - In Exercises 1-18, s=1+2i,u=32i,v=4+i,w=2i, and...Ch. 4.6 - Prob. 2ECh. 4.6 - In Exercises 1-18, s=1+2i,u=32i,v=4+i,w=2i, and...Ch. 4.6 - In Exercises 1-18, s=1+2i,u=32i,v=4+i,w=2i, and...Ch. 4.6 - In Exercises 1-18, s=1+2i,u=32i,v=4+i,w=2i, and...Ch. 4.6 - In Exercises 1-18, s=1+2i,u=32i,v=4+i,w=2i, and...Ch. 4.6 - In Exercises 1-18, s=1+2i,u=32i,v=4+i,w=2i, and...Ch. 4.6 - In Exercises 1-18, s=1+2i,u=32i,v=4+i,w=2i, and...Ch. 4.6 - In Exercises 1-18, s=1+2i,u=32i,v=4+i,w=2i, and...Ch. 4.6 - In Exercises 1-18, s=1+2i,u=32i,v=4+i,w=2i, and...Ch. 4.6 - In Exercises 1-18, s=1+2i,u=32i,v=4+i,w=2i, and...Ch. 4.6 - In Exercises 1-18, s=1+2i,u=32i,v=4+i,w=2i, and...Ch. 4.6 - In Exercises 1-18, s=1+2i,u=32i,v=4+i,w=2i, and...Ch. 4.6 - In Exercises 1-18, s=1+2i,u=32i,v=4+i,w=2i, and...Ch. 4.6 - In Exercises 1-18, s=1+2i,u=32i,v=4+i,w=2i, and...Ch. 4.6 - In Exercises 1-18, s=1+2i,u=32i,v=4+i,w=2i, and...Ch. 4.6 - In Exercises 1-18, s=1+2i,u=32i,v=4+i,w=2i, and...Ch. 4.6 - Prob. 18ECh. 4.6 - Find the eigenvalues and the eigenvectors for the...Ch. 4.6 - Find the eigenvalues and the eigenvectors for the...Ch. 4.6 - Find the eigenvalues and the eigenvectors for the...Ch. 4.6 - Find the eigenvalues and the eigenvectors for the...Ch. 4.6 - Find the eigenvalues and the eigenvectors for the...Ch. 4.6 - Find the eigenvalues and the eigenvectors for the...Ch. 4.6 - In Exercises 25 and 26, solve the linear system....Ch. 4.6 - In Exercises 25 and 26, solve the linear system....Ch. 4.6 - In Exercises 27-30, calculate x. x=[1+i2]Ch. 4.6 - In Exercises 27-30, calculate x. x=[3+i2i]Ch. 4.6 - In Exercises 27-30, calculate x. x=[12ii3+i]Ch. 4.6 - In Exercises 27-30, calculate x. x=[2i1i3]Ch. 4.6 - Prob. 31ECh. 4.6 - In Exercises 31-34, use linear algebra software to...Ch. 4.6 - Prob. 33ECh. 4.6 - Prob. 34ECh. 4.6 - Establish the five properties of the conjugate...Ch. 4.6 - Let A be an (mn) matrix, and let B be an (np)...Ch. 4.6 - Prob. 37ECh. 4.6 - An (nn) matrix A is called Hermitian if A*=A....Ch. 4.6 - Let p(t)=a0+a1t+...+antn, where the coefficients...Ch. 4.6 - Prob. 40ECh. 4.6 - A real symmetric (nn) matrix A is called positive...Ch. 4.6 - An (nn) matrix A is called unitary if A*A=I. If A...Ch. 4.7 - In Exercises 1 12, determine whether the given...Ch. 4.7 - In Exercises 1 12, determine whether the given...Ch. 4.7 - In Exercises 1 12, determine whether the given...Ch. 4.7 - In Exercises 1 12, determine whether the given...Ch. 4.7 - In Exercises 1 12, determine whether the given...Ch. 4.7 - In Exercises 1 12, determine whether the given...Ch. 4.7 - In Exercises 1 12, determine whether the given...Ch. 4.7 - In Exercises 1 12, determine whether the given...Ch. 4.7 - In Exercises 1 12, determine whether the given...Ch. 4.7 - In Exercises 1 12, determine whether the given...Ch. 4.7 - In Exercises 1 12, determine whether the given...Ch. 4.7 - In Exercises 1 12, determine whether the given...Ch. 4.7 - In Exercises 13 18, use condition 5 to determine...Ch. 4.7 - In Exercises 13 18, use condition 5 to determine...Ch. 4.7 - In Exercises 13 18, use condition 5 to determine...Ch. 4.7 - In Exercises 13 18, use condition 5 to determine...Ch. 4.7 - In Exercises 13 18, use condition 5 to determine...Ch. 4.7 - In Exercises 13 18, use condition 5 to determine...Ch. 4.7 - In Exercises 19 and 20, find values ,,a,bandc such...Ch. 4.7 - In Exercises 19 and 20, find values ,,a,bandc such...Ch. 4.7 - Let A be an (nn) matrix, and let S be a...Ch. 4.7 - Show that if A is diagonalizable and if B is...Ch. 4.7 - Suppose that B is similar to A. Show each of the...Ch. 4.7 - Prove properties b and c of Theorem 21. Hint: For...Ch. 4.7 - Let u be a vector in Rn such that uTu=1. Let...Ch. 4.7 - Suppose that A and B are orthogonal (nn) matrices....Ch. 4.7 - Prob. 31ECh. 4.7 - Prob. 32ECh. 4.7 - Prob. 33ECh. 4.7 - Prob. 34ECh. 4.7 - Prob. 35ECh. 4.7 - Prob. 36ECh. 4.7 - Prob. 37ECh. 4.7 - Prob. 38ECh. 4.7 - Let B=QTAQ, where q and A are as in Exercise 38....Ch. 4.7 - Prob. 40ECh. 4.7 - Following the outline of Exercises 38-40, use...Ch. 4.7 - Consider the (nn) symmetric matrix A=(aij) defined...Ch. 4.7 - Suppose that A is a real symmetric matrix and that...Ch. 4.8 - In Exercises 1-6, consider the vector sequence...Ch. 4.8 - Prob. 2ECh. 4.8 - In Exercises 1-6, consider the vector sequence...Ch. 4.8 - Prob. 4ECh. 4.8 - In Exercises 1-6, consider the vector sequence...Ch. 4.8 - Prob. 6ECh. 4.8 - In Exercises 7-14, let xk=Axk1, k=1,2,....... for...Ch. 4.8 - Prob. 8ECh. 4.8 - In Exercises 7-14, let xk=Axk1, k=1,2,....... for...Ch. 4.8 - Prob. 10ECh. 4.8 - In Exercises 7-14, let xk=Axk1, k=1,2,, for the...Ch. 4.8 - Prob. 12ECh. 4.8 - Prob. 13ECh. 4.8 - Prob. 14ECh. 4.8 - Prob. 15ECh. 4.8 - In Exercises 15-18, solve the initial-value...Ch. 4.8 - Prob. 17ECh. 4.8 - Prob. 18ECh. 4.8 - Prob. 19ECh. 4.8 - Prob. 20ECh. 4.8 - Prob. 21ECh. 4.8 - Prob. 22ECh. 4.8 - Prob. 23ECh. 4.8 - Prob. 24ECh. 4.8 - Prob. 25ECh. 4.8 - Prob. 26ECh. 4.8 - Prob. 27ECh. 4.8 - Prob. 28ECh. 4.8 - Prob. 29ECh. 4.SE - Prob. 1SECh. 4.SE - Prob. 2SECh. 4.SE - Prob. 3SECh. 4.SE - Prob. 4SECh. 4.SE - Prob. 5SECh. 4.SE - Prob. 6SECh. 4.SE - Prob. 7SECh. 4.SE - Prob. 8SECh. 4.SE - Prob. 9SECh. 4.SE - Prob. 10SECh. 4.SE - Prob. 11SECh. 4.SE - Prob. 12SECh. 4.SE - Prob. 13SECh. 4.SE - Prob. 14SECh. 4.CE - CONCEPTUAL EXERCISES In Exercises 18, answer true...Ch. 4.CE - Prob. 2CECh. 4.CE - CONCEPTUAL EXERCISES In Exercises 18, answer true...Ch. 4.CE - Prob. 4CECh. 4.CE - Prob. 5CECh. 4.CE - Prob. 6CECh. 4.CE - Prob. 7CECh. 4.CE - CONCEPTUAL EXERCISES In Exercises 18, answer true...Ch. 4.CE - Prob. 9CECh. 4.CE - In Exercises 9-14, give a brief answer. Suppose...Ch. 4.CE - In Exercises 9-14, give a brief answer. Show that...Ch. 4.CE - In Exercises 9-14, give a brief answer. Let A and...Ch. 4.CE - Prob. 13CECh. 4.CE - In Exercises 9-14, give a brief answer. Let u be a...
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