WEBVTT

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Here is the problem: Find the eigenvalues of the matrix [5 4; 1 2].

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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 6: solve (A minus λI)v equals 0.

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Eigenvectors for λ equals 1: solve (A minus λI)v equals 0.

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So the answer is λ equals 6, 1.

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The eigenvalues are the roots of the characteristic polynomial. Eigenvectors: λ equals 6: ⟨4, 1⟩; λ equals 1: ⟨-1, 1⟩.

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Sum of eigenvalues approximately 7 equals the trace 7.

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Product of eigenvalues equals the determinant 6.