Find the Eigenvalues [[3,0,0],[-3,9,18],[2,-3,-6]]

Problem

[[3,0,0],[−3,9,18],[2,−3,−6]]

Solution

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

det(A−λ*I)=0

2. Write the determinant expression for the given matrix.

|[3−λ,0,0],[−3,9−λ,18],[2,−3,−6−λ]|=0

3. Expand the determinant along the first row, since it contains two zeros.

(3−λ)*|[9−λ,18],[−3,−6−λ]|=0

4. Calculate the 2×2 determinant inside the expression.

(3−λ)*((9−λ)*(−6−λ)−(18)*(−3))=0

5. Simplify the quadratic expression inside the parentheses.

(3−λ)*(−54−9*λ+6*λ+λ2+54)=0

6. Combine like terms to find the characteristic polynomial.

(3−λ)*(λ2−3*λ)=0

7. Factor the quadratic part of the equation.

(3−λ)*λ*(λ−3)=0

8. Solve for λ by setting each factor to zero.

(λ_1)=3

(λ_2)=0

(λ_3)=3

Final Answer

λ=0,3,3

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