WEBVTT

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Here is the problem: Find the eigenvalues of the matrix [2 1 0; 1 2 0; 0 0 3].

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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 3: solve (A minus λI)v equals 0. Algebraic multiplicity 2, geometric multiplicity 2.

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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 3, 3, 1.

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The eigenvalues are the roots of the characteristic polynomial. Eigenvectors: λ equals 3: ⟨1, 1, 0⟩, ⟨0, 0, 1⟩; λ equals 1: ⟨-1, 1, 0⟩.

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Sum of eigenvalues approximately 7 equals the trace 7.

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Product of eigenvalues equals the determinant 9.