Find the Eigenvalues [[4,-6,-2],[2,-9,-4],[-4,2,-2]]

Problem

[[4,−6,−2],[2,−9,−4],[−4,2,−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.

|A−λ*I|=|[4−λ,−6,−2],[2,−9−λ,−4],[−4,2,−2−λ]|=0

2. Expand the determinant along the first row.

(4−λ)*|[−9−λ,−4],[2,−2−λ]|−(−6)*|[2,−4],[−4,−2−λ]|+(−2)*|[2,−9−λ],[−4,2]|=0

3. Calculate the 2×2 determinants.

(4−λ)*[(−9−λ)*(−2−λ)−(−8)]+6*[2*(−2−λ)−16]−2*[4−(−4)*(−9−λ)]=0

4. Simplify the expressions inside the brackets.

(4−λ)*[λ2+11*λ+18+8]+6*[−4−2*λ−16]−2*[4−(36+4*λ)]=0

(4−λ)*(λ2+11*λ+26)+6*(−2*λ−20)−2*(−4*λ−32)=0

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

(4*λ2+44*λ+104−λ3−11*λ2−26*λ)−12*λ−120+8*λ+64=0

−λ3−7*λ2+14*λ+48=0

6. Multiply by −1 to simplify the polynomial.

λ3+7*λ2−14*λ−48=0

7. Factor the cubic equation. By testing small integers, we find λ=−2 is a root since (−2)3+7*(−2)2−14*(−2)−48=−8+28+28−48=0

(λ+2)*(λ2+5*λ−24)=0

8. Factor the remaining quadratic expression.

(λ+2)*(λ+8)*(λ−3)=0

9. Solve for λ to find the eigenvalues.

(λ_1)=−2,(λ_2)=−8,(λ_3)=3

Final Answer

λ=−8,−2,3

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