Which of the following could be the set of distinct eigenvalues for a real 3×33×3 matrix? Select all that apply.{−2,2,5}{−2,2,5}{4−3i,5i,3i+4}{4−3i,5i,3i+4}{−3,1}{−3,1}{−2i,2i}{−2i,2i}{−2,3,−2i−4}{−2,3,−2i−4}{−2,4−3i,3i+4}{−2,4−3i,3i+4}{−5,1−6i,6i−1}
Question
Which of the following could be the set of distinct eigenvalues for a real 3×33×3 matrix? Select all that apply.{−2,2,5}{−2,2,5}{4−3i,5i,3i+4}{4−3i,5i,3i+4}{−3,1}{−3,1}{−2i,2i}{−2i,2i}{−2,3,−2i−4}{−2,3,−2i−4}{−2,4−3i,3i+4}{−2,4−3i,3i+4}{−5,1−6i,6i−1}
Solution
The eigenvalues of a real matrix are either real or come in complex conjugate pairs. This is because the characteristic polynomial of a matrix has real coefficients, and the complex roots of such polynomials always come in conjugate pairs.
Given this, let's evaluate each set:
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{−2,2,5}: This set is valid because all the eigenvalues are real.
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{4−3i,5i,3i+4}: This set is not valid because 5i does not have a conjugate pair.
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{−3,1}: This set is not valid because a 3x3 matrix should have 3 eigenvalues.
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{−2i,2i}: This set is not valid because a 3x3 matrix should have 3 eigenvalues.
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{−2,3,−2i−4}: This set is not valid because −2i−4 does not have a conjugate pair.
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{−2,4−3i,3i+4}: This set is valid because 4−3i and 3i+4 are conjugate pairs and -2 is a real number.
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{−5,1−6i,6i−1}: This set is valid because 1−6i and 6i−1 are conjugate pairs and -5 is a real number.
So, the sets of distinct eigenvalues for a real 3x3 matrix could be {−2,2,5}, {−2,4−3i,3i+4}, and {−5,1−6i,6i−1}.
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