WEBVTT

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Here is the problem: Find the eigenvalues of the matrix [6 minus 2; 2 1].

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Find the eigenvalues and eigenvectors of a 2 times 2 matrix.

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Set up the characteristic equation.

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For a 2 times 2 matrix it is λ squared minus (trace)λ plus det.

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Characteristic polynomial.

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Rational roots found with the rational root theorem.

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Eigenvectors for λ equals 5: solve (A minus λI)v equals 0.

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Eigenvectors for λ equals 2: solve (A minus λI)v equals 0.

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So the answer is λ equals 5, 2.

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The eigenvalues are the roots of the characteristic polynomial. Eigenvectors: λ equals 5: ⟨2, 1⟩; λ equals 2: ⟨1, 2⟩.

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Sum of eigenvalues approximately 7 equals the trace 7.

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Product of eigenvalues equals the determinant 10.