Find the Eigenvalues

Problem

det([[4,2,3,3],[0,2,h,3],[0,0,4,14],[0,0,0,2]]−λ*I)=0

Solution

1. Identify the matrix type. The given matrix is an upper triangular matrix because all entries below the main diagonal are zero.

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

3. Set up the characteristic equation. The eigenvalues λ are the solutions to the equation det(A−λ*I)=0

4. Write the characteristic polynomial. For this upper triangular matrix, the determinant is the product of the diagonal elements minus λ

(4−λ)*(2−λ)*(4−λ)*(2−λ)=0

5. Solve for λ Setting each factor to zero gives the eigenvalues.

(4−λ)2*(2−λ)2=0

6. List the distinct values. The solutions are λ=4 and λ=2 each with an algebraic multiplicity of 2.

Final Answer

λ={4,2}

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