MATH 225 Certification Exam Questions and CORRECT Answers

If the columns of A are linearly dependent Then the matrix is not invertible and an eigenvalue is 0 Note that A−1 exists. In order for λ−1 to be an eigenvalue of A−1, there must exist a nonzero x such that Upper A Superscript negative 1 Baseline Bold x equals lambda Superscript negative 1 Baseline Bold x . A−1x=λ−1x. Suppose a nonzero x satisfies Ax=λx. What is the first operation that should be performed on Ax=λx so that an equation similar to the one in the previous step can be obtained? Left-multiply both sides of Ax=λx by A−1. Show that if A2 is the zero matrix, then the only eigenvalue of A is 0. If Ax=λx for some x≠0, then 0x=A2x=A(Ax)=A(λx)=λAx=λ2x=0. Since x is nonzero, λ must be zero. Thus, each eigenvalue of A is zero.