Find the Eigenvalues [[6,0,0],[-5,-13,0],[9,7,4]]

Problem

[[6,0,0],[−5,−13,0],[9,7,4]]

Solution

1. Identify the matrix A and the characteristic equation det(A−λ*I)=0 where I is the identity matrix and λ represents the eigenvalues.

A=[[6,0,0],[−5,−13,0],[9,7,4]]

2. Set up the determinant for the characteristic equation by subtracting λ from the diagonal elements of the matrix.

det(6−λ)=0

3. Observe that the matrix is a lower triangular matrix, meaning all entries above the main diagonal are zero.

det(A−λ*I)=(6−λ)*(−13−λ)*(4−λ)

4. Solve the characteristic equation by setting each factor equal to zero to find the values of λ

(6−λ)*(−13−λ)*(4−λ)=0

5. Extract the roots, which are the eigenvalues of the matrix.

(λ_1)=6

(λ_2)=−13

(λ_3)=4

Final Answer

λ=6,−13,4

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