Math 225 Final Exam questions with answers.

Math 225 Final Exam questions with |\ |\ |\ |\ |\ |\




answers


If the columns of A are linearly dependent - CORRECT ANSWERS
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✔✔Then the matrix is not invertible and an eigenvalue is 0
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Note that A−1 exists. In order for λ−1 to be an eigenvalue of
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A−1, there must exist a nonzero x such that Upper A Superscript
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negative 1 Baseline Bold x equals lambda Superscript negative 1
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Baseline Bold x . A−1x=λ−1x. Suppose a nonzero x satisfies
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Ax=λx. What is the first operation that should be performed on
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Ax=λx so that an equation similar to the one in the previous step
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can be obtained? - CORRECT ANSWERS ✔✔Left-multiply both
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sides of Ax=λx by A−1. |\ |\ |\ |\




Show that if A2 is the zero matrix, then the only eigenvalue of A
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is 0. - CORRECT ANSWERS ✔✔If Ax=λx for some x≠0, then
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0x=A2x=A(Ax)=A(λx)=λAx=λ2x=0. Since x is nonzero, λ must |\ |\ |\ |\ |\ |\ |\


be zero. Thus, each eigenvalue of A is zero.
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Finding the characteristic polynomial of a 3 x 3 matrix - CORRECT
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ANSWERS ✔✔Add the first two columns to the right side of the
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matrix and then add the down diagonals and subtract the up
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diagonals


In a simplified n x n matrix the Eigenvalues are - CORRECT
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ANSWERS ✔✔The values of the main diagonal |\ |\ |\ |\ |\ |\

, Use a property of determinants to show that A and AT have the
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same characteristic polynomial - CORRECT ANSWERS ✔✔Start
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with detAT−λI)=detAT−λI)=det(A−λI)T. Then use the formula det
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AT=det A. |\




The determinant of A is the product of the diagonal entries in A.
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Select the correct choice below and, if necessary, fill in the
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answer box to complete your choice. - CORRECT ANSWERS
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✔✔The statement is false because the determinant of the
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2×2 matrix A= [ 1 1 (1 1 below) ] is not equal to the product of
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the entries on the main diagonal of A.
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An elementary row operation on A does not change the
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determinant. Choose the correct answer below. - CORRECT |\ |\ |\ |\ |\ |\ |\ |\


ANSWERS ✔✔The statement is false because scaling a row also
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scales the determinant by the same scalar factor.
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(det A)(det B)=detAB. Select the correct choice below and, if
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necessary, fill in the answer box to complete your choice. -|\ |\ |\ |\ |\ |\ |\ |\ |\ |\ |\


CORRECT ANSWERS ✔✔The statement is true because it is the
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Multiplicative Property of determinants. |\ |\ |\




If λ+5 is a factor of the characteristic polynomial of A, then 5 is
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an eigenvalue of A. Select the correct choice below and, if
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necessary, fill in the answer box to complete your choice. -|\ |\ |\ |\ |\ |\ |\ |\ |\ |\ |\


CORRECT ANSWERS ✔✔The statement is false because in order
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for 5 to be an eigenvalue of A, the characteristic polynomial
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would need to have a factor of λ−5.
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