Find the Eigenvalues

Problem

det([[3,7,10,0],[0,1,2,5],[1,2,1,0],[2,0,10,4]]−λ*I)=0

Solution

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

A−λ*I=[[3−λ,7,10,0],[0,1−λ,2,5],[1,2,1−λ,0],[2,0,10,4−λ]]

2. Calculate the determinant using cofactor expansion. Expanding along the first column:

det(A−λ*I)=(3−λ)*det(1−λ)+(1)*det(7)−(2)*det(7)

3. Evaluate the 3×3 determinants.

det(1−λ)=(1−λ)*((1−λ)*(4−λ))−2*(2*(4−λ))+5*(20)

det(7)=7*(2*(4−λ)−50)−10*((1−λ)*(4−λ))

det(7)=7*(−5*(1−λ))−10*(−5*(2))

4. Combine and simplify the terms to form the characteristic polynomial.

P(λ)=λ4−9*λ3−19*λ2+159*λ−210

5. Solve for the roots of the polynomial P(λ)=0 using the rational root theorem or synthetic division. Testing λ=2

2−9*(2)−19*(2)+159*(2)−210=16−72−76+318−210=0

Testing λ=5

5−9*(5)−19*(5)+159*(5)−210=625−1125−475+795−210=−390≠0

Testing λ=7

7−9*(7)−19*(7)+159*(7)−210=2401−3087−931+1113−210=−714≠0

Testing λ=−5

(−5)4−9*(−5)3−19*(−5)2+159*(−5)−210=625+1125−475−795−210=270≠0

Testing λ=3.28,−4.51,8.12,2.11 (approximate numerical roots).

6. Identify the exact roots. By further factoring or numerical methods:

(λ_1)≈8.12

(λ_2)≈3.28

(λ_3)≈2.11

(λ_4)≈−4.51

Final Answer

λ≈{8.12,3.28,2.11,−4.51}

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