Find the Eigenvalues [[2,2,1],[1,3,1],[1,2,2]]

Problem

[[2,2,1],[1,3,1],[1,2,2]]

Solution

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

det(A−λ*I)=0

|[2−λ,2,1],[1,3−λ,1],[1,2,2−λ]|=0

2. Expand the determinant along the first row.

(2−λ)*|[3−λ,1],[2,2−λ]|−2*|[1,1],[1,2−λ]|+1*|[1,3−λ],[1,2]|=0

3. Calculate the 2×2 determinants inside the expansion.

(2−λ)*((3−λ)*(2−λ)−2)−2*((2−λ)−1)+1*(2−(3−λ))=0

4. Simplify the expressions within the brackets.

(2−λ)*(λ2−5*λ+4)−2*(1−λ)+(−1+λ)=0

5. Distribute and combine like terms to form the characteristic polynomial.

(2*λ2−10*λ+8−λ3+5*λ2−4*λ)−2+2*λ−1+λ=0

−λ3+7*λ2−11*λ+5=0

6. Factor the polynomial by testing possible roots. Testing λ=1 gives −1+7−11+5=0 so (λ−1) is a factor.

−(λ−1)*(λ2−6*λ+5)=0

7. Factor the remaining quadratic expression.

−(λ−1)*(λ−1)*(λ−5)=0

−(λ−1)2*(λ−5)=0

8. Solve for λ to find the eigenvalues.

(λ_1)=1

(λ_2)=1

(λ_3)=5

Final Answer

λ=1,1,5

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