1. For the following questions first indicate if it\'s True or False then explai
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1. For the following questions first indicate if it's True or False then explain why citing relevant rules or theorems from the textbook. If Vi and v2 are linearly independent eigenvectors, then they correspond to distinct eigenva- T F lues T F The eigenvalues of a matrix are on its main diagonal T F A number c is an eigenvalue of A if and only if the equation (cI - A)x 0 has a nontrivial solution. T F A matrix A is not invertible if and only if 0 is an eigenvalue of A. T F IA + 5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A. T F The multiplicity of a root r of the characteristic equation of A is called the algebraic mu plicity of r as an eigenvalue of A T F A is diagonalizable if A has n eigenvectors.Explanation / Answer
1a. The statement that “If v1 and v2 are linearly independent eigenvectors, then they correspond to different eigenvalues “ is FALSE. The converse is True.
b. The statement that “The eigenvalues of a matrix are on its main diagonal” is False. The statement is true only for a triamgular matrix.
c. The statement that “A number c is an eigenvalue of A if and only if the equation (cI-A)x = 0 has a non-trivial solutiion” is True. The equation (cI-A)x = 0 is same as Ax = cx.
d. The statement that “A matrix A is not invertible if and only if 0 is an eigenvalue of A’’ is True. The characteristic equation of a matrix A is det(A-I) = 0. If = 0, then it changes to det(A) = 0 , in which case, A is not invertible.
e. The statement that “If +5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A” is False as -5 is an eigenvalue of A.
f. The statement that” The multiplicity of a root r of the characteristic equation of a matrix A is called the algebraic multiplicity of the eigenvalue r is True.( it is the definituion of algebraic multiplicity of an eigenvalue).
g. The statement that ”A is diagonalizable if A has n eigenvectors” is False. Firstly the size of A has not been mentioned here and secondly, A should have n distinct and linearly independent eigenvectors in order to be diagonalizable.
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