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

Problem

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

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,1,1],[1,2−λ,1],[1,1,2−λ]|=0

2. Expand the determinant along the first row to create a polynomial in terms of λ

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

3. Simplify the sub-determinants using the formula a*d−b*c

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

(2−λ)*(λ2−4*λ+3)−(1−λ)+(λ−1)=0

4. Combine terms to form the characteristic polynomial.

(2−λ)*(λ−1)*(λ−3)+2*(λ−1)=0

5. Factor out the common term (λ−1) to solve for the roots.

(λ−1)*((2−λ)*(λ−3)+2)=0

(λ−1)*(−λ2+5*λ−6+2)=0

(λ−1)*(−λ2+5*λ−4)=0

6. Factor the remaining quadratic to find all eigenvalues.

−(λ−1)*(λ2−5*λ+4)=0

−(λ−1)*(λ−1)*(λ−4)=0

Final Answer

λ=1,1,4

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