True-False Review
For Questions (a)-(h), decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem in fact from the text. If false, provide an example, illustration, or brief explanation of why the statement is false.
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Differential Equations and Linear Algebra (4th Edition)
- CAPSTONE Explain how to determine whether an nn matrix A is diagonalizable using a similar matrices, b eigenvectors, and c distinct eigenvalues.arrow_forwardGuide Proof Prove nonzero nilpotent matrices are not diagonalizable Getting started: From Exercises 80 in Section 7.1, you know that 0 is the only eigenvalue of the nilpotent matrix A. Show that it is impossible for A to be diagonalizable. i Assume A is diagonalizable, so there exists an invertible matrix P such that P1AP=D, here D is the zero matrix. ii Find A in term of P,P1 and D. iii Find a contradiction and conclude that nonzero nilpotent matrices are not diagonalizable.arrow_forwardTrue or False? In Exercises 67 and 68, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a Geometrically, if is an eigenvalue of a matrix A and x is an eigenvector of A corresponding to , then multiplying x by A produce a vector x parallel to x. b If A is nn matrix with an eigenvalue , then the set of all eigenvectors of is a subspace of Rn.arrow_forward
- For what values of a does the matrix A=[01a1] have the characteristics below? a A has eigenvalue of multiplicity 2. b A has 1 and 2 as eigenvalues. c A has real eigenvalues.arrow_forwardDiagonalizing a Matrix In Exercise 7-14, find if possible a nonsingular matrix P such that P1AP is diagonal. Verify thatP1APis a diagonal matrix with the eigenvalues on the main diagonal A=[6321] See Exercise 15, section 7.1. Characteristic Equation, Eigenvalues, and Eigenvectors in Exercise 15-28, find a the characteristics equation and b the eigenvalues and corresponding eigenvectors of the matrix. [6321]arrow_forwardDiagonalizable Matrices and Eigenvalues In Exercise 1-6, a verify that A is diagonalizable by finding P1AP, and b use the result of part a and Theorem 7.4 to find the eigenvalues of A. A=[1136310],P=[3411]arrow_forward
- True or False? In Exercises 37 and 38, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a If A and B are similar nn matrix, then they have always the same characteristics polynomial equation. b The fact that an nn matrix A has n distinct eigenvalues does not guarantee that A is diagonalizable.arrow_forwardTrue or False? In Exercises 69 and 70, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a An eigenvalue of a matrix A is a scalar such that det(IA)=0. b An eigenvector may be the zero vector 0. c A matrix A is orthogonally diagonalizable when there exists an orthogonal matrix P such that P1AP=D is diagonal.arrow_forward
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