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+λ2+45*e(−2*x))−3*(36*e(−6*x)−27*e(−6*x)+3*e(−4*x)*λ)=0

(6*e(−4*x)−λ)*(λ2+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)*λ2+18*e(−4*x)*λ−54*e(−6*x)*λ+108*e(−6*x)−λ3−3*λ2+9*e(−2*x)*λ2−18*e(−2*x)*λ−27*e(−6*x)−9*e(−4*x)*λ=0

−λ3+λ2*(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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