A is an nxn matrix. Mark each statement below True or False. Justify each answer.a. If Ax = Ax for some scalar A, then x is an eigenvector ofA. Choose the correct answer below.O A. True. If Ax = ix for some scalar A, then x is an eigenvector of A because the only solution to this equation is the trivial solution.O B. False. The condition that Ax = ix for some scalar i is not sufficient to determine if x is an eigenvector of A. The vector x must be nonzero.O c. True. If Ax = ix for some scalar A, then x is an eigenvector of A because A is an inverse of A.O D. False. The equation Ax = Ax is not useddetermine eigenvectors. If Ax = 0 for some scalar i, then x is an eigenvector of A.b. If v, and v, are linearly independent eigenvectors, then they correspond to distinct eigenvalues. Choose the correct answer below.O A. False. Every eigenvalue has an infinite number of corresponding eigenvectors.O B. False. There may be linearly independent eigenvectors that both correspond to the same eigenvalue.O c. True. If v, and v, are linearly independent eigenvectors, then they correspond to distinct eigenvalues, because eigenvectors must be nonzero.O D. True. If v, and v, are linearly independent eigenvectors, then they correspond to distinct eigenvalues, because all eigenvectors of a single eigenvalue are linearly dependent.c. A steady-state vector for a stochastic matrixactually an eigenvector. Choose the correct answer below.O A. True. A steady-state vector for a stochastic matrix is actually an eigenvector because it satisfies the equation Ax = x.O B. False. A steady-state vector for a stochastic matrix is not an eigenvector because it does not satisfy the equation Ax = 0.O c. False. A steady-state vector for a stochastic matrix is not an eigenvector because it does not satisfy the equation Ax = xO D. True. A steady-state vector for a stochastic matrix is actually an eigenvector because it satisfies the equation Ax = 0.d. The eigenvalues of a matrix are on its main diagonal. Choose the correct answer below.O A. False. The matrix must first be reduced to echelon form. The eigenvalues are on the main diagonal of the echelon form of the matrix.O B. True. The eigenvalues of a matrix are on its main diagonal because the main diagonal determines the pivots of the matrix, which are used to calculate the eigenvalues.O c. True. The eigenvalues of a matrix are on its main diagonal because the main diagonal remains the same when the matrix is transposed, and a matrix and its transpose have the same eigenvalues.O D. False. If the matrix is a triangular matrix, the values on the main diagonal are eigenvalues. Otherwise, the main diagonal may or may not contain eigenvalues. e. An eigenspace of A is a null space of a certain matrix. Choose the correct answer below.O A. True. An eigenspace of A corresponding to the eigenvalue is the null space of the matrix (AA - 1).O B. True. An eigenspace of A corresponding to the eigenvalue i is the null space of the matrix (A -A1).O c. False. An eigenspace of A is not a null space of a certain matrix because an eigenspace consists of all solutions x to the equation Ax = Ab, which does not include the zero vector unless b=0.O D. False. An eigenspace of A is not a null space of a certain matrix because an eigenspace consists of all the eigenvectors corresponding to an eigenvalue A, and eigenvectors are by definition nonzero vectors, so the eigenspace does not include the zero vector.

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A is an nxn matrix. Mark each statement below True or False. Justify each answer. a. If Ax = Ax for some scalar A, then x is an eigenvector of A. Choose the correct answer below. A. True. If Ax = x for some scalar A, then x is an eigenvector of A because the only solution to this equation is the trivial solution. B. False. The condition that Ax = Ax for some scalar A is not sufficient to determine if x is an eigenvector of A. The vector x must be nonzero. C. True. If Ax = x for some scalar A, then x is an eigenvector of A because . is an inverse of A. D. False. The equation Ax = Ax is not used to determine eigenvectors. If Ax = 0 for some scalar A, then x is an eigenvector of A.

A is an nxn matrix. Mark each statement below True or False. Justify each answer. a. If Ax = Ax for some scalar A, then x is an eigenvector of A. Choose the correct answer below. A. True. If Ax = x for some scalar A, then x is an eigenvector of A because the only solution to this equation is the trivial solution. B. False. The condition that Ax = Ax for some scalar A is not sufficient to determine if x is an eigenvector of A. The vector x must be nonzero. C. True. If Ax = x for some scalar A, then x is an eigenvector of A because . is an inverse of A. D. False. The equation Ax = Ax is not used to determine eigenvectors. If Ax = 0 for some scalar A, then x is an eigenvector of A.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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