Find the Eigenvalues A=[[0,1,0],[0,0,1],[729k^3,-243k^2,27k]]

Problem

A=[[0,1,0],[0,0,1],[729*k3,−243*k2,27*k]]

Solution

1. Set up the characteristic equation by finding the determinant of A−λ*I

det(A−λ*I)=det(−λ)=0

2. Expand the determinant along the first row.

−λ*det(−λ)−1*det(0)=0

3. Calculate the 2×2 determinants.

−λ*(−λ*(27*k−λ)−(−243*k2))−1*(0*(27*k−λ)−729*k3)=0

4. Simplify the resulting polynomial expression.

−λ*(λ2−27*k*λ+243*k2)+729*k3=0

−λ3+27*k*λ2−243*k2*λ+729*k3=0

5. Multiply by −1 to standardize the polynomial.

λ3−27*k*λ2+243*k2*λ−729*k3=0

6. Recognize the pattern of a binomial expansion (a−b)3=a3−3*a2*b+3*a*b2−b3

λ3−3*(λ2)*(9*k)+3*(λ)*(9*k)2−(9*k)3=0

7. Factor the cubic equation.

(λ−9*k)3=0

8. Solve for λ

λ=9*k

Final Answer

λ=9*k,9*k,9*k

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