Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
9th Edition
ISBN: 9780321962218
Author: Steven J. Leon
Publisher: PEARSON
expand_more
expand_more
format_list_bulleted
Textbook Question
Chapter 6.3, Problem 13E
Show that a nonzero nilpotent matrix is defective.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionChapter 6 Solutions
Linear Algebra with Applications (9th Edition) (Featured Titles for Linear Algebra (Introductory))
Ch. 6.1 - Find the eigenvalues and the corresponding...Ch. 6.1 - Show that the eigenvalues of a triangular matrix...Ch. 6.1 - Let A be an nn matrix. Prove that A is singular if...Ch. 6.1 - Let A be a nonsingular matrix and let be an...Ch. 6.1 - Let A and B be nn matrices. Show that if none of...Ch. 6.1 - Let be an eigenvalue of A and let x be an...Ch. 6.1 - Let A bean nn matrix and let B=I2A+A2. Show that...Ch. 6.1 - An nn matrix A is said to be idempotent if A2=A....Ch. 6.1 - An nn matrix is said to be nilpotent if Ak=O for...Ch. 6.1 - Let A be an nn matrix and let B=AI for some scalar...
Ch. 6.1 - Let A be an nn matrix and let B=A+I. Is it...Ch. 6.1 - Show that A and AT have the same eigenvalues. Do...Ch. 6.1 - Show that the matrix A=( cos sin sin cos) will...Ch. 6.1 - Let A be a 22 matrix. If tr(A)=8 and det(A)=12,...Ch. 6.1 - Let A=(aij) be an nn matrix with eigenvalues...Ch. 6.1 - Prob. 16ECh. 6.1 - Prob. 17ECh. 6.1 - Let A be an nn matrix and let be an eigenvalue of...Ch. 6.1 - Prob. 19ECh. 6.1 - Let =+bi and =c+di be complex scalars and let A...Ch. 6.1 - Let Q be an orthogonal matrix. Show that if is an...Ch. 6.1 - Let Q be an orthogonal matrix with an eigenvalue...Ch. 6.1 - Let Q be a 33 orthogonal matrix whose determinant...Ch. 6.1 - Let x1,...,xr be eigenvectors of an nn matrix A...Ch. 6.1 - Let A bean nn matrix and let be an eigenvalue of...Ch. 6.1 - Let B=S1AS and let x be an eigenvector of B...Ch. 6.1 - Let A be an nn matrix with an eigenvalue and let...Ch. 6.1 - Prob. 28ECh. 6.1 - Let A be an nn matrix and let be a nonzero...Ch. 6.1 - Prob. 30ECh. 6.1 - Let A be a matrix whose columns all add up to a...Ch. 6.1 - Let 1 and 2 be distinct eigenvalues of A. Let x be...Ch. 6.1 - Let A and B be nn matrices. Show that (a) If is a...Ch. 6.1 - Prove that there do not exist nn matrices A and B...Ch. 6.1 - Let p()=(1)n(nan1n1a1a0) be a polynomial of degree...Ch. 6.1 - The result given in Exercise 35(b) holds even if...Ch. 6.2 - Find the general solution of each of the following...Ch. 6.2 - Solve each of the following initial value...Ch. 6.2 - Given Y=c1e1tx1+c2e2tx2++cnentxn is the solution...Ch. 6.2 - Two tanks each contain 100 liters of a mixture....Ch. 6.2 - Prob. 5ECh. 6.2 - Solve the initial value problem...Ch. 6.2 - In Application 2, assume that the solutions are of...Ch. 6.2 - Solve the the problem in Application 2, using the...Ch. 6.2 - Prob. 9ECh. 6.2 - Three masses are connected by a series of springs...Ch. 6.2 - Transform the nth-order equation...Ch. 6.3 - In each of the following, factor the matrix A into...Ch. 6.3 - For each of the matrices in Exercise 1, use the...Ch. 6.3 - For each of the nonsingular matrices in Exercise...Ch. 6.3 - For each of the following, find a matrix B such...Ch. 6.3 - Let A be a nondefective nn matrix with...Ch. 6.3 - Let A be a diagonalizable matrix whose eigenvalues...Ch. 6.3 - Show that any 33 matrix of the form (a100a100b) is...Ch. 6.3 - For each of the following, find all possible...Ch. 6.3 - Let A be a 44 matrix and let be an eigenvalue of...Ch. 6.3 - Prob. 10ECh. 6.3 - Let A be a nn matrix with real entries and let...Ch. 6.3 - Let A be an nn matrix with an eigenvalue of...Ch. 6.3 - Show that a nonzero nilpotent matrix is defective.Ch. 6.3 - Let A be a diagonalizable matrix and let X be the...Ch. 6.3 - It follows from Exercise 14 that for a...Ch. 6.3 - Prob. 16ECh. 6.3 - Let x, y, be nonzero vectors in n,n2, and let...Ch. 6.3 - Let A be a diagonalizable nn matrix. Prove that if...Ch. 6.3 - Prob. 19ECh. 6.3 - Let T be an upper triangular matrix with distinct...Ch. 6.3 - Each year, employees at a company are given the...Ch. 6.3 - The city of Mawtookit maintains a constant...Ch. 6.3 - Let A=( 1 2 1 3 1 5 1 4 1 3 2 5 1 4 1 3 2 5 ) be a...Ch. 6.3 - Prob. 24ECh. 6.3 - Prob. 25ECh. 6.3 - Consider a Web network consisting of only four...Ch. 6.3 - Prob. 27ECh. 6.3 - The transition matrix in Example 5 has the...Ch. 6.3 - Let A be the PageRank transition matrix and let xk...Ch. 6.3 - Use the definition of the matrix exponential to...Ch. 6.3 - Compute eA for each of the following matrices: A=(...Ch. 6.3 - In each of the following, solve the initial value...Ch. 6.3 - Let X be an eigenvalue of an nn matrix A and let x...Ch. 6.3 - Show that eA is nonsingular for any diagonalizable...Ch. 6.3 - Let A be a diagonalizable matrix with...Ch. 6.4 - For each of the following pairs of vectors z and...Ch. 6.4 - Let z1=( 1+i 2 1i 2 ) and z2=( i 2 1 2 ) Show...Ch. 6.4 - Let {u1,u2} be an orthonormal basis for 2, and let...Ch. 6.4 - Which of the matrices that follow are Hermitian?...Ch. 6.4 - Find an orthogonal or unitary diagonalizing matrix...Ch. 6.4 - Prob. 6ECh. 6.4 - Let A be an nn Hermitian matrix and let x be a...Ch. 6.4 - Let A be an Hermitian matrix and let B=iA. Show...Ch. 6.4 - Let A and C be matrices in mn and let Bnr. Prove...Ch. 6.4 - Prob. 10ECh. 6.4 - Show that z,w=wHz defines an inner product on n.Ch. 6.4 - Let x, y, and z be vectors in n and let and be...Ch. 6.4 - Let {u1,...,un} be an orthonormal basis for a...Ch. 6.4 - Given that A=(40001i0 i1) find a matrix B such...Ch. 6.4 - Let U be a unitary matrix. Prove that U is normal....Ch. 6.4 - Let u be a unit vector in n and define U=I2uuH....Ch. 6.4 - Show that if a matrix U is both unitary and...Ch. 6.4 - Let A be a 22 matrix with Schur decomposition UTUH...Ch. 6.4 - Let A be a 55 matrix with real entries. Let A=QTQT...Ch. 6.4 - Let A be a nn matrix with Schur decomposition...Ch. 6.4 - Show that M=A+iB (where A and B are real matrices)...Ch. 6.4 - Prob. 22ECh. 6.4 - Prob. 23ECh. 6.4 - Prob. 24ECh. 6.4 - Prob. 25ECh. 6.4 - Let A be a Hermitian matrix with eigenvalues...Ch. 6.4 - Let A=(0110) Write A as a sum 1u1u1T+2u2u2T, where...Ch. 6.4 - Let A be a Hermitian matrix with eigenvalues 12n...Ch. 6.4 - Given Amm,Bnn,Cmn, the equation AXXB=C(3) is known...Ch. 6.5 - Show that A and AT have the same nonzero singular...Ch. 6.5 - Use the method of Example 1 to find the singular...Ch. 6.5 - For each of the matrices in Exercise 2: determine...Ch. 6.5 - Let A=( 28 20 14 19 102 21)=( 3 5 4 5 0 4 5 3 5...Ch. 6.5 - The matrix A=(254630630254) has singular value...Ch. 6.5 - Prove that if A is a symmetric matrix with...Ch. 6.5 - Let A be an mn matrix with singular value...Ch. 6.5 - Let A be an nn matrix. Show that ATA and AAT are...Ch. 6.5 - Let A be an nn matrix with singular values...Ch. 6.5 - Let A be an nn matrix with singular value...Ch. 6.5 - Show that if is a singular value of A then there...Ch. 6.5 - Let A be an mn matrix of rank n with singular...Ch. 6.5 - Prob. 13ECh. 6.6 - Find the matrix associated with each of the...Ch. 6.6 - Reorder the eigenvalues in Example 2 so that 1=4...Ch. 6.6 - Prob. 3ECh. 6.6 - Let 1 and 2 be the eigenvalues of A=(abbc) What...Ch. 6.6 - Prob. 5ECh. 6.6 - Which of the matrices that follow are positive...Ch. 6.6 - For each of the following functions, determine...Ch. 6.6 - Show that if A is symmetric positive definite,...Ch. 6.6 - Prob. 9ECh. 6.6 - Prob. 10ECh. 6.6 - Let A be a symmetric nn matrix with eigenvalues...Ch. 6.6 - Prob. 12ECh. 6.6 - Prob. 13ECh. 6.6 - Let A be a symmetric positive definite nn matrix....Ch. 6.7 - For each of the following matrices, compute the...Ch. 6.7 - Prob. 2ECh. 6.7 - Let A=(2 100 12 100 12 100 12) Compute the LU...Ch. 6.7 - For each of the following, factor the given matrix...Ch. 6.7 - Find the Cholesky decomposition LLT for each of...Ch. 6.7 - Prob. 6ECh. 6.7 - Prove each of the following: If U is a unit upper...Ch. 6.7 - Prob. 8ECh. 6.7 - Let A be a symmetric positive definite matrix with...Ch. 6.7 - Let A be an mn matrix with rank n. Show that the...Ch. 6.7 - Prob. 11ECh. 6.7 - Let A be a symmetric positive definite matrix and...Ch. 6.7 - Prob. 13ECh. 6.7 - Prob. 14ECh. 6.7 - Prob. 15ECh. 6.7 - Let A be an nn symmetric negative definite matrix....Ch. 6.7 - Prob. 17ECh. 6.8 - Find the eigenvalues of each of the following...Ch. 6.8 - Prob. 2ECh. 6.8 - Find the output vector x in the open version of...Ch. 6.8 - Consider the closed version of the Leontief...Ch. 6.8 - Prob. 5ECh. 6.8 - Prob. 6ECh. 6.8 - Which of the matrices that follow are reducible?...Ch. 6.8 - Prob. 8ECh. 6.8 - Prob. 9ECh. 6.8 - Prove that a 22 matrix A is reducible if and only...Ch. 6.8 - Prove the Forbenius theorem in the case where A is...Ch. 6.8 - Prob. 12ECh. 6.8 - Let A be an nn positive stochastic matrix with...Ch. 6.8 - Would the results of parts (c) and (d) in Exercise...Ch. 6.8 - A management student received fellowship offers...Ch. 6 - The top matrix on the menu is the diagonal matrix...Ch. 6 - Prob. 2ECh. 6 - Prob. 3ECh. 6 - Prob. 4ECh. 6 - Prob. 5ECh. 6 - Prob. 6ECh. 6 - Prob. 7ECh. 6 - The last item on the eigshow menu will generate a...Ch. 6 - Prob. 9ECh. 6 - Prob. 10ECh. 6 - Prob. 11ECh. 6 - Consider the matrices A=(5 33 5) and B=(5 335)...Ch. 6 - Prob. 13ECh. 6 - Prob. 14ECh. 6 - Prob. 15ECh. 6 - Prob. 16ECh. 6 - Prob. 17ECh. 6 - Prob. 18ECh. 6 - Prob. 19ECh. 6 - Let A be a nonsingular 22 matrix with singular...Ch. 6 - Set A=[1,1;0.5,0.5] and use MATLAB to verify each...Ch. 6 - Prob. 22ECh. 6 - Prob. 23ECh. 6 - Prob. 24ECh. 6 - Prob. 25ECh. 6 - If A is an nn matrix whose eigenvalues are all...Ch. 6 - If A is nn matrix, then A and AT have the same...Ch. 6 - If A and B are similar matrices, then they have...Ch. 6 - If A and B are nn matrices with the same...Ch. 6 - If A has eigenvalues of multiplicity greater than...Ch. 6 - If A is a 44 matrix of rank 3 and =0 is an...Ch. 6 - If A is a 44 matrix of rank 1 and =0 is an...Ch. 6 - The rank of an nn matrix A is equal to the number...Ch. 6 - The rank of an mn matrix A is equal to the number...Ch. 6 - If A is Hermitian and c is a complex scalar, then...Ch. 6 - If an nn matrix A has Schur decomposition A=UTUH,...Ch. 6 - If A is normal, but not Hermitian, then A must...Ch. 6 - Prob. 13CTACh. 6 - Prob. 14CTACh. 6 - If A is symmetric, then eA is symmetric positive...Ch. 6 - Let A=(10011 112 2) Find the eigenvalues of A. For...Ch. 6 - Let A be a 44 matrix with real entries that has...Ch. 6 - Let A be a nonsingular nn matrix and let be an...Ch. 6 - Show that if A is a matrix of the form...Ch. 6 - Let A=(4222 10 102 10 14) Without computing the...Ch. 6 - Prob. 6CTBCh. 6 - Prob. 7CTBCh. 6 - Let A be a 44 real symmetric matrix with...Ch. 6 - Let {u1,u2} be an orthonormal basis for 2 and...Ch. 6 - Let A be a 55 nonsymmetric matrix with rank equal...Ch. 6 - Let A and B be nn matrices. If A is real and...Ch. 6 - Let A be a matrix whose singular value...
Additional Math Textbook Solutions
Find more solutions based on key concepts
The given equation.
High School Math 2015 Common Core Algebra 1 Student Edition Grade 8/9
In Exercises 1-6, consider the numbers 23, 6, 3, 2.45, 266, 18.4, 11, 273, 516, 7.151551555 , 35, 35,87, 0, 16....
College Algebra: Graphs and Models (6th Edition)
The inverse of the function f(x)=x3 and then graph the function and its inverse on the same pair of coordinate ...
Algebra and Trigonometry: Structure and Method, Book 2
Factor each expression completely. 41.28y2+43y48
High School Math 2012 Common-core Algebra 1 Practice And Problem Solvingworkbook Grade 8/9
The polynomial of degree 3 and zeros as −2, 3, 5.
College Algebra (5th Edition)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Similar questions
- Show that no 22 matrices A and B exist that satisfy the matrix equation. AB-BA=1001.arrow_forwardShow that the matrix below does not have an LU factorization. A=0110arrow_forwardCan a matrix whose entry is 0 on the diagonal be solved? Explain why or why not. What would you do to remedy the situation?arrow_forward
- Can a matrix with an entire column of zeros have an inverse? Explain why or why not.arrow_forwardExplain what it means in terms of an inverse for a matrix to have a 0 determinant.arrow_forwardIn general, it is difficult to show that two matrices are similar. However, if two similar matrices are diagonalizable, the task becomes easier. In Exercises 38-41, show that A and B are similar by showing that they are similar to the same diagonal matrix. Then find an invertible matrix P such that .arrow_forward
- Determine if the statement is true or false. If the statement is false, then correct it and make it true. For the product of two matrices to be defined, the number of rows of the first matrix must equal the number of columns of the second matrix.arrow_forwardCan a matrix that has 0 entries for an entire row have one solution? Explain why or why not.arrow_forward
arrow_back_ios
arrow_forward_ios
Recommended textbooks for you
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningElements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageCollege Algebra (MindTap Course List)AlgebraISBN:9781305652231Author:R. David Gustafson, Jeff HughesPublisher:Cengage LearningLinear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
Publisher:Cengage Learning
Elements Of Modern Algebra
Algebra
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Cengage Learning,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Finite Math: Markov Chain Example - The Gambler's Ruin; Author: Brandon Foltz;https://www.youtube.com/watch?v=afIhgiHVnj0;License: Standard YouTube License, CC-BY
Introduction: MARKOV PROCESS And MARKOV CHAINS // Short Lecture // Linear Algebra; Author: AfterMath;https://www.youtube.com/watch?v=qK-PUTuUSpw;License: Standard Youtube License
Stochastic process and Markov Chain Model | Transition Probability Matrix (TPM); Author: Dr. Harish Garg;https://www.youtube.com/watch?v=sb4jo4P4ZLI;License: Standard YouTube License, CC-BY