Find the Eigenvalues [[6,8,9],[x,8,6],[5,6,5]]

Problem

det([[6,8,9],[x,8,6],[5,6,5]]−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(6−λ)=0

2. Expand the determinant along the first row.

(6−λ)*det(8−λ)−8*det(x)+9*det(x)=0

3. Calculate the 2×2 determinants.

(6−λ)*((8−λ)*(5−λ)−36)−8*(x*(5−λ)−30)+9*(6*x−5*(8−λ))=0

4. Simplify the expressions inside the parentheses.

(6−λ)*(λ2−13*λ+4)−8*(5*x−x*λ−30)+9*(6*x−40+5*λ)=0

5. Distribute the terms to form a cubic polynomial in λ

(6*λ2−78*λ+24−λ3+13*λ2−4*λ)−(40*x−8*x*λ−240)+(54*x−360+45*λ)=0

6. Combine like terms to find the characteristic polynomial.

−λ3+19*λ2+(8*x−37)*λ+(14*x−96)=0

7. Multiply by −1 to standardize the polynomial.

λ3−19*λ2+(37−8*x)*λ+(96−14*x)=0

Final Answer

λ3−19*λ2+(37−8*x)*λ+(96−14*x)=0

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