Find the Eigenvalues

Problem

[[31/6,−73/18,−16/9],[−73/18,91/18,16/9],[−16/9,16/9,2]]

Solution

1. Set up the characteristic equation by subtracting λ from the diagonal elements and setting the determinant of the resulting matrix A−λ*I to zero.

det(A−λ*I)=0

2. Calculate the determinant of the matrix.

|[31/6−λ,−73/18,−16/9],[−73/18,91/18−λ,16/9],[−16/9,16/9,2−λ]|=0

3. Expand the determinant to find the characteristic polynomial.

−λ3+110/9*λ2−251/9*λ+10=0

4. Multiply by −9 to simplify the polynomial equation.

9*λ3−110*λ2+251*λ−90=0

5. Factor the cubic polynomial to find the roots.

(9*λ−4)*(λ−9)*(λ−5/2)=0

(9*λ−4)*(λ−9)*(2*λ−5)=0

6. Solve for λ to identify the eigenvalues.

(λ_1)=9

(λ_2)=5/2

(λ_3)=4/9

Final Answer

λ=9,5/2,4/9

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