Find the Eigenvalues [[-2,0,-36],[0,-3,0],[-36,0,-23]]

Problem

det([[−2,0,−36],[0,−3,0],[−36,0,−23]]−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(−2−λ)=0

2. Expand the determinant along the second row or second column, since it contains two zeros.

(−3−λ)*det(−2−λ)=0

3. Calculate the determinant of the remaining 2×2 matrix.

(−3−λ)*[(−2−λ)*(−23−λ)−(−36)*(−36)]=0

4. Simplify the expression inside the brackets.

(−3−λ)*[46+2*λ+23*λ+λ2−1296]=0

(−3−λ)*[λ2+25*λ−1250]=0

5. Factor the quadratic expression λ2+25*λ−1250

(−3−λ)*(λ+50)*(λ−25)=0

6. Solve for λ by setting each factor equal to zero.

−3−λ=0⇒(λ_1)=−3

λ+50=0⇒(λ_2)=−50

λ−25=0⇒(λ_3)=25

Final Answer

λ=−3,−50,25

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