Find the Eigenvalues [[2,8,-20],[2,2,-10],[2,4,-12]]

Problem

[[2,8,−20],[2,2,−10],[2,4,−12]]

Solution

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

det(A−λ*I)=0

2. Write the determinant expression for the matrix.

|[2−λ,8,−20],[2,2−λ,−10],[2,4,−12−λ]|=0

3. Expand the determinant along the first row.

(2−λ)*|[2−λ,−10],[4,−12−λ]|−8*|[2,−10],[2,−12−λ]|−20*|[2,2−λ],[2,4]|=0

4. Calculate the 2×2 determinants.

(2−λ)*((2−λ)*(−12−λ)+40)−8*(2*(−12−λ)+20)−20*(8−2*(2−λ))=0

5. Simplify the internal algebraic expressions.

(2−λ)*(λ2+10*λ+16)−8*(−2*λ−4)−20*(2*λ+4)=0

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

−λ3−8*λ2+4*λ+32+16*λ+32−40*λ−80=0

−λ3−8*λ2−20*λ−16=0

7. Multiply by −1 to simplify the polynomial for factoring.

λ3+8*λ2+20*λ+16=0

8. Factor the polynomial. By testing small integers, λ=−2 is a root. Using synthetic division or grouping, the polynomial factors into (λ+2)*(λ2+6*λ+8)

(λ+2)*(λ+2)*(λ+4)=0

9. Solve for the roots of the equation to find the eigenvalues.

(λ_1)=−2,(λ_2)=−2,(λ_3)=−4

Final Answer

λ=−2,−2,−4

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