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

Problem

det([[0,1,0],[0,0,1],[729*k^3,−243*k^2,27*k]]−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(−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*k^2))−(0*(27*k−λ)−729*k^3)=0

4. Simplify the resulting polynomial expression.

−λ*(λ^2−27*k*λ+243*k^2)+729*k^3=0

−λ^3+27*k*λ^2−243*k^2*λ+729*k^3=0

5. Multiply by −1 to standardize the polynomial.

λ^3−27*k*λ^2+243*k^2*λ−729*k^3=0

6. Recognize the expression as a perfect cube binomial expansion of the form (a−b)^3=a^3−3*a^2*b+3*a*b^2−b^3

(λ−9*k)^3=0

7. Solve for λ by taking the cube root of both sides.

λ−9*k=0

λ=9*k

Final Answer

λ=9*k,9*k,9*k

Want more problems? Check here!