Find the Eigenvalues [[4,0,1],[2,3,2],[49,0,4]]

Problem

det([[4,0,1],[2,3,2],[49,0,4]]−λ*I)=0

Solution

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

det(4−λ)=0

2. Expand the determinant along the second column, which contains two zeros, to simplify the calculation.

(3−λ)⋅det(4−λ)=0

3. Calculate the determinant of the remaining 2×2 matrix.

(3−λ)*((4−λ)*(4−λ)−(1)*(49))=0

4. Simplify the expression inside the parentheses by expanding the quadratic and combining like terms.

(3−λ)*(λ2−8*λ+16−49)=0

(3−λ)*(λ2−8*λ−33)=0

5. Factor the quadratic expression λ2−8*λ−33 into linear factors.

(3−λ)*(λ−11)*(λ+3)=0

6. Solve for λ by setting each factor equal to zero.

(λ_1)=3

(λ_2)=11

(λ_3)=−3

Final Answer

λ=3,11,−3

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