WEBVTT

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Here is the problem: Find the eigenvalues of the matrix [2 0 0; 0 3 4; 0 4 9].

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Find the eigenvalues and eigenvectors of a 3 times 3 matrix.

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Set up the characteristic equation.

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

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So the answer is λ equals 11, 2, 1.

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The eigenvalues are the roots of the characteristic polynomial. Eigenvectors: λ equals 11: ⟨0, 1, 2⟩; λ equals 2: ⟨1, 0, 0⟩; λ equals 1: ⟨0, negative 2, 1⟩.

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Sum of eigenvalues approximately 14 equals the trace 14.

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Product of eigenvalues equals the determinant 22.