Find the Eigenvalues [[0,1],[1,0]]

Problem

[[0,1],[1,0]]

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.

det(A−λ*I)=0

2. Write the determinant for the given 2×2 matrix.

|[0−λ,1],[1,0−λ]|=0

3. Expand the determinant using the formula a*d−b*c

(−λ)*(−λ)−(1)*(1)=0

4. Simplify the characteristic polynomial to a standard quadratic form.

λ2−1=0

5. Solve for λ by factoring the difference of squares or moving the constant to the other side.

λ2=1

6. Extract the square roots to find the two distinct eigenvalues.

λ=±1

Final Answer

λ=1,−1

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