True 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
(b) If A is
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Elementary Linear Algebra (MindTap Course List)
- True 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_forwardTrue 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 is a diagonalizable matrix, then it has n linearly independent eigenvectors. b If an nn matrix A is diagonalizable, then it must have n distinct eigenvalues.arrow_forwardDetermine a Sufficient Condition for Diagonalization In Exercises 23-26, find the eigenvalues of the matrix and determine there is a sufficient number of eigenvalues to guarantee that the matrix is diagonalizable by Theorem 7.6. [2052]arrow_forward
- Diagonalizable 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=[110030425],P=[013040122]arrow_forwardCAPSTONE Explain how to determine whether an nn matrix A is diagonalizable using a similar matrices, b eigenvectors, and c distinct eigenvalues.arrow_forwardDiagonalizing a Matrix In Exercise 7-14, find if possible a nonsingular matrix P such that P1AP is diagonal. Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal A=[122252663] See Exercise 23, section 7.1. Characteristic Equation, Eigenvalues, and EigenvectorsIn Exercise 15-28, find a the characteristics equation and b the eigenvalues and corresponding eigenvectors of the matrix. [122252663]arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning