Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)
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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Chapter 4.7, Problem 3E

In Exercises 1 – 12, determine whether the given matrix A is diagonalizable. If A is diagonalizable, calculate A 5 using the method of Example 2.

A = [ − 3 2 − 2 1 ]

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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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