Find the Eigenvalues [[7,0,1],[-8,1,0],[-8,0,1]]

Problem

[[7,0,1],[−8,1,0],[−8,0,1]]

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(A−λ*I)=0

2. Write the determinant expression for the given matrix.

|[7−λ,0,1],[−8,1−λ,0],[−8,0,1−λ]|=0

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

(1−λ)*|[7−λ,1],[−8,1−λ]|=0

4. Calculate the 2×2 determinant inside the expression.

(1−λ)*((7−λ)*(1−λ)−(1)*(−8))=0

5. Simplify the quadratic expression inside the parentheses.

(1−λ)*(7−7*λ−λ+λ^2+8)=0

(1−λ)*(λ^2−8*λ+15)=0

6. Factor the quadratic polynomial λ^2−8*λ+15

(1−λ)*(λ−3)*(λ−5)=0

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

1−λ=0⇒λ=1

λ−3=0⇒λ=3

λ−5=0⇒λ=5

Final Answer

λ=1,3,5

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