Find the Eigenvalues [[1,2,3],[4,5,6],[7,8,9]]

Problem

det([[1,2,3],[4,5,6],[7,8,9]]−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(1−λ)=0

2. Expand the determinant along the first row.

(1−λ)*((5−λ)*(9−λ)−48)−2*(4*(9−λ)−42)+3*(32−7*(5−λ))=0

3. Simplify the terms inside the parentheses.

(1−λ)*(λ2−14*λ−3)−2*(−4*λ−6)+3*(7*λ−3)=0

4. Distribute and combine like terms to form the characteristic polynomial.

−λ3+15*λ2+18*λ=0

5. Factor the polynomial to find the roots.

−λ*(λ2−15*λ−18)=0

6. Solve for λ using the zero product property and the quadratic formula for the remaining quadratic factor.

(λ_1)=0

λ=(15±√(,225−4*(1)*(−18)))/2

λ=(15±√(,297))/2

λ=(15±3√(,33))/2

Final Answer

λ=0,(15+3√(,33))/2,(15−3√(,33))/2

Want more problems? Check here!