Find the Eigenvalues

Problem

[[1,0,0,1],[0,1,1,0],[0,1,1,0],[1,0,0,3]]

Solution

1. Set up the characteristic equation by finding the determinant of A−λ*I

det(1−λ)=0

2. Expand the determinant using cofactor expansion along the first row.

(1−λ)*det(1−λ)−1*det(0)=0

3. Calculate the 3×3 determinants.

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

4. Factor out the common term ((1−λ)2−1)

((1−λ)2−1)*((1−λ)*(3−λ)−1)=0

5. Simplify the first factor.

(1−2*λ+λ2−1)=λ2−2*λ=λ*(λ−2)

6. Simplify the second factor.

(λ2−4*λ+3−1)=λ2−4*λ+2

7. Solve for the roots of the characteristic polynomial.

λ*(λ−2)*(λ2−4*λ+2)=0

8. Apply the quadratic formula to λ2−4*λ+2=0

λ=(4±√(,16−8))/2=2±√(,2)

Final Answer

λ=0,2,2+√(,2),2−√(,2)

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