Find the Eigenvalues

Problem

[[6*e^(−4*x),0,−3],[12*e^(−4*x),9*e^(−2*x),−15],[3*e^(−4*x),3*e^(−2*x),−3]]

Solution

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

det(A−λ*I)=0

2. Write the determinant expression for the matrix.

|[6*e^(−4*x)−λ,0,−3],[12*e^(−4*x),9*e^(−2*x)−λ,−15],[3*e^(−4*x),3*e^(−2*x),−3−λ]|=0

3. Expand the determinant along the first row to simplify the calculation.

(6*e^(−4*x)−λ)*|[9*e^(−2*x)−λ,−15],[3*e^(−2*x),−3−λ]|−0+(−3)*|[12*e^(−4*x),9*e^(−2*x)−λ],[3*e^(−4*x),3*e^(−2*x)]|=0

4. Calculate the 2×2 determinants.

(6*e^(−4*x)−λ)*((9*e^(−2*x)−λ)*(−3−λ)−(−15)*(3*e^(−2*x)))−3*(12*e^(−4*x)*(3*e^(−2*x))−3*e^(−4*x)*(9*e^(−2*x)−λ))=0

5. Simplify the terms inside the parentheses.

(6*e^(−4*x)−λ)*(−27*e^(−2*x)−9*e^(−2*x)*λ+3*λ+λ^2+45*e^(−2*x))−3*(36*e^(−6*x)−27*e^(−6*x)+3*e^(−4*x)*λ)=0

(6*e^(−4*x)−λ)*(λ^2+λ*(3−9*e^(−2*x))+18*e^(−2*x))−3*(9*e^(−6*x)+3*e^(−4*x)*λ)=0

6. Distribute and combine like terms to form the cubic polynomial.

6*e^(−4*x)*λ^2+18*e^(−4*x)*λ−54*e^(−6*x)*λ+108*e^(−6*x)−λ^3−3*λ^2+9*e^(−2*x)*λ^2−18*e^(−2*x)*λ−27*e^(−6*x)−9*e^(−4*x)*λ=0

−λ^3+λ^2*(6*e^(−4*x)−3+9*e^(−2*x))+λ*(9*e^(−4*x)−54*e^(−6*x)−18*e^(−2*x))+81*e^(−6*x)=0

7. Identify the roots of the characteristic equation. By testing values or factoring, we find the values of λ that satisfy the equation.

(λ_1)=3*e^(−2*x)

(λ_2)=6*e^(−4*x)

(λ_3)=6*e^(−2*x)−3

Final Answer

λ=3*e^(−2*x),6*e^(−4*x),6*e^(−2*x)−3

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