Find the Eigenvalues [[-4,0,1],[3,-6,3],[1,0,-4]]

Problem

[[−4,0,1],[3,−6,3],[1,0,−4]]

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.

det(−4−λ)=0

2. Expand the determinant along the second column, which contains two zeros, to simplify the calculation.

(−6−λ)*det(−4−λ)=0

3. Calculate the determinant of the 2×2 submatrix.

(−6−λ)*((−4−λ)*(−4−λ)−(1)*(1))=0

4. Simplify the expression inside the parentheses.

(−6−λ)*((λ+4)2−1)=0

5. Expand the quadratic term.

(−6−λ)*(λ2+8*λ+16−1)=0

(−6−λ)*(λ2+8*λ+15)=0

6. Factor the quadratic expression.

−(λ+6)*(λ+3)*(λ+5)=0

7. Solve for λ by setting each factor to zero.

(λ_1)=−6

(λ_2)=−3

(λ_3)=−5

Final Answer

λ=−6,−5,−3

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