Find the Eigenvalues [[2,2,1],[x,-9,0],[1,2,1]]

Problem

Eigenvalues of *[[2,2,1],[0,−9,0],[1,2,1]]

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(2−λ)=0

2. Expand the determinant along the second row, as it contains the most zeros.

(−9−λ)*det(2−λ)=0

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

(−9−λ)*((2−λ)*(1−λ)−(1)*(1))=0

4. Simplify the quadratic expression within the parentheses.

(−9−λ)*(λ2−3*λ+2−1)=0

(−9−λ)*(λ2−3*λ+1)=0

5. Solve for the first eigenvalue by setting the first factor to zero.

−9−λ=0⇒(λ_1)=−9

6. Apply the quadratic formula to solve λ2−3*λ+1=0 for the remaining eigenvalues.

λ=(−(−3)±√(,(−3)2−4*(1)*(1)))/(2*(1))

λ=(3±√(,9−4))/2

λ=(3±√(,5))/2

Final Answer

λ=−9,(3+√(,5))/2,(3−√(,5))/2

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