Find the Eigenvalues [[8,0,0],[-9,13,0],[5,8,1]]

Problem

[[8,0,0],[−9,13,0],[5,8,1]]

Solution

1. Identify the type of matrix. The given matrix is a lower triangular matrix because all entries above the main diagonal are zero.

2. Apply the property of triangular matrices. For any triangular matrix (upper or lower), the eigenvalues are simply the entries located on the main diagonal.

3. Extract the diagonal elements. The elements on the main diagonal are 8 13 and 1

4. Verify using the characteristic equation. The eigenvalues λ are the roots of det(A−λ*I)=0 For this matrix, the determinant is (8−λ)*(13−λ)*(1−λ)=0

Final Answer

λ=8,13,1

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