Find the Eigenvalues [[4,0,1],[2,3,2],[1,0,4]]

Problem

[[4,0,1],[2,3,2],[1,0,4]]

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

|[4−λ,0,1],[2,3−λ,2],[1,0,4−λ]|=0

2. Expand the determinant along the second column, as it contains two zeros, making the calculation simpler.

(3−λ)*|[4−λ,1],[1,4−λ]|=0

3. Calculate the 2×2 determinant by multiplying the diagonal elements and subtracting the product of the off-diagonal elements.

(3−λ)*((4−λ)*(4−λ)−(1)*(1))=0

(3−λ)*((4−λ)2−1)=0

4. Simplify the expression inside the parentheses using the difference of squares or by expanding the quadratic.

(3−λ)*(16−8*λ+λ2−1)=0

(3−λ)*(λ2−8*λ+15)=0

5. Factor the quadratic expression to find all possible values for λ

(3−λ)*(λ−3)*(λ−5)=0

−(3−λ)2*(λ−5)=0

6. Solve for λ by setting each factor equal to zero.

(λ_1)=3

(λ_2)=3

(λ_3)=5

Final Answer

λ=3,5

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