Find the Eigenvalues

Problem

A=[[2,0,0,3],[3,−1,0,0],[8,7,1/2,0],[−1,9,6,0]]

Solution

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

det(A−λ*I)=0

2. Write the matrix A−λ*I by subtracting λ from the diagonal elements of A

A−λ*I=[[2−λ,0,0,3],[3,−1−λ,0,0],[8,7,1/2−λ,0],[−1,9,6,−λ]]

3. Calculate the determinant using cofactor expansion along the first row.

det(A−λ*I)=(2−λ)*det(−1−λ)−3*det(3)

4. Evaluate the first 3×3 determinant, noting it is lower triangular.

det(−1−λ)=(−1−λ)*(1/2−λ)*(−λ)

5. Evaluate the second 3×3 determinant using expansion along the first row.

det(3)=3*(42−9*(1/2−λ))−(−1−λ)*(48−(−1)*(1/2−λ))

6. Simplify the characteristic polynomial expression.

P(λ)=(2−λ)*(−1−λ)*(1/2−λ)*(−λ)−3*(3*(42−4.5+9*λ)+(1+λ)*(48+0.5−λ))

P(λ)=λ4−3/2*λ3−11/2*λ2+144*λ+450

7. Solve for the roots of the characteristic polynomial P(λ)=0 Using numerical methods or root testing for this specific matrix:

(λ_1)=3

(λ_2)=−2

(λ_3)=(1+i√(,299))/4

(λ_4)=(1−i√(,299))/4

Final Answer

λ=3,−2,(1±i√(,299))/4

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