Find the Eigenvalues [[-2,0,0],[0,-2,0],[0,0,13]]

Problem

[[−2,0,0],[0,−2,0],[0,0,13]]

Solution

1. Identify the matrix A as a diagonal matrix.

A=[[−2,0,0],[0,−2,0],[0,0,13]]

2. Recall the property that for any diagonal matrix, the eigenvalues are the entries located on the main diagonal.

det(A−λ*I)=0

3. Set up the characteristic equation by subtracting λ from the diagonal elements and setting the determinant to zero.

|[−2−λ,0,0],[0,−2−λ,0],[0,0,13−λ]|=0

4. Calculate the determinant of the diagonal matrix, which is the product of its diagonal entries.

(−2−λ)*(−2−λ)*(13−λ)=0

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

−2−λ=0⇒λ=−2

13−λ=0⇒λ=13

Final Answer

λ=−2,−2,13

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