Find the Eigenvalues

Problem

det(4−λ)=0

Solution

1. Set up the characteristic equation by subtracting λ from the diagonal elements of the matrix A and setting the determinant of A−λ*I to zero.

2. Expand the determinant of the 4×4 matrix using cofactor expansion or row reduction to find the characteristic polynomial.

3. Calculate the characteristic polynomial, which for this matrix is P(λ)=λ4−20*λ3−15*λ2+370*λ−293

4. Solve for the roots of the polynomial P(λ)=0 to find the eigenvalues.

5. Approximate the roots using numerical methods, as the quartic equation does not yield simple integers.

Final Answer

λ≈{18.334,−5.395,6.166,0.895}

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