Find the Eigenvalues [[5,7,-5],[0,4,-1],[2,8,-3]]

Problem

[[5,7,−5],[0,4,−1],[2,8,−3]]

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

|[5−λ,7,−5],[0,4−λ,−1],[2,8,−3−λ]|=0

2. Expand the determinant along the first column to simplify the calculation.

(5−λ)*|[4−λ,−1],[8,−3−λ]|−0+2*|[7,−5],[4−λ,−1]|=0

3. Calculate the 2×2 determinants inside the expansion.

(5−λ)*((4−λ)*(−3−λ)−(−8))+2*(−7−(−5*(4−λ)))=0

(5−λ)*(λ2−λ−12+8)+2*(−7+20−5*λ)=0

4. Simplify the resulting polynomial expression.

(5−λ)*(λ2−λ−4)+2*(13−5*λ)=0

(5*λ2−5*λ−20−λ3+λ2+4*λ)+(26−10*λ)=0

−λ3+6*λ2−11*λ+6=0

5. Solve the characteristic equation by multiplying by −1 and factoring the cubic polynomial.

λ3−6*λ2+11*λ−6=0

(λ−1)*(λ−2)*(λ−3)=0

6. Identify the roots of the polynomial, which are the eigenvalues of the matrix.

(λ_1)=1

(λ_2)=2

(λ_3)=3

Final Answer

λ=1,2,3

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