Find the Eigenvalues [[0.4,1-c],[0.6,c]]

Problem

[[0.4,1−c],[0.6,c]]

Solution

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

det(0.4−λ)=0

2. Expand the determinant using the formula (a−λ)*(d−λ)−b*c

(0.4−λ)*(c−λ)−(0.6)*(1−c)=0

3. Distribute the terms to form a quadratic equation in terms of λ

0.4*c−0.4*λ−c*λ+λ2−(0.6−0.6*c)=0

4. Combine like terms and rearrange into standard quadratic form λ2+b*λ+c=0

λ2−(0.4+c)*λ+(c−0.6)=0

5. Factor the quadratic equation or use the quadratic formula to solve for λ

(λ−1)*(λ−(c−0.6))=0

6. Identify the roots of the equation, which are the eigenvalues.

(λ_1)=1

(λ_2)=c−0.6

Final Answer

λ=1,c−0.6

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