Find the Eigenvalues

Problem

[[−5,0,0,0],[0,−5,5,5],[5,0,−2,−2],[−5,0,2,2]]

Solution

1. Set up the characteristic equation by finding the determinant of A−λ*I where I is the identity matrix.

det(A−λ*I)=|[−5−λ,0,0,0],[0,−5−λ,5,5],[5,0,−2−λ,−2],[−5,0,2,2−λ]|=0

2. Expand the determinant along the first row, since it contains three zeros.

det(A−λ*I)=(−5−λ)*|[−5−λ,5,5],[0,−2−λ,−2],[0,2,2−λ]|

3. Expand the resulting 3×3 determinant along its first column.

det(A−λ*I)=(−5−λ)*(−5−λ)*|[−2−λ,−2],[2,2−λ]|

4. Calculate the 2×2 determinant using the formula a*d−b*c

|[−2−λ,−2],[2,2−λ]|=(−2−λ)*(2−λ)−(−2)*(2)

⇒(−2−λ)*(2−λ)+4=(−4+2*λ−2*λ+λ2)+4

⇒λ2−4+4=λ2

5. Combine the factors to form the characteristic polynomial.

(−5−λ)2*λ2=0

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

λ+5=0⇒λ=−5

λ2=0⇒λ=0

Final Answer

λ=−5,−5,0,0

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