Find the Eigenvalues

Problem

A=[[1,0,1,1],[0,1,1,1],[1,1,1,0],[1,1,0,1]]

Solution

1. Set up the characteristic equation by finding the determinant of A−λ*I

det(A−λ*I)=det(1−λ)=0

2. Simplify the determinant using row operations. Subtract the second row from the first row.

det(1−λ)=0

3. Factor out (1−λ) from the first row.

(1−λ)*det(1)=0

4. Perform further row operations. Add the first column to the second column.

(1−λ)*det(1)=0

5. Expand along the first row to reduce the matrix to a 3×3

(1−λ)*det(1−λ)=0

6. Calculate the 3×3 determinant.

(1−λ)*((1−λ)*((1−λ)2−0)−1*(2*(1−λ)−0)+1*(0−2*(1−λ)))=0

7. Simplify the resulting polynomial expression.

(1−λ)*((1−λ)3−2*(1−λ)−2*(1−λ))=0

(1−λ)*((1−λ)3−4*(1−λ))=0

8. Factor the expression completely.

(1−λ)2*((1−λ)2−4)=0

(1−λ)2*(1−λ−2)*(1−λ+2)=0

(1−λ)2*(−1−λ)*(3−λ)=0

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

λ=1,λ=1,λ=−1,λ=3

Final Answer

λ={1,1,−1,3}

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