Find the Eigenvalues [[1,4],[1,-2]]

Problem

[[1,4],[1,−2]]

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 2×2 matrix.

|[1−λ,4],[1,−2−λ]|=0

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

(1−λ)*(−2−λ)−(4)*(1)=0

4. Simplify the quadratic equation by distributing the terms and combining like terms.

−2−λ+2*λ+λ2−4=0

λ2+λ−6=0

5. Factor the quadratic equation to find the roots.

(λ+3)*(λ−2)=0

6. Solve for the eigenvalues by setting each factor to zero.

(λ_1)=−3

(λ_2)=2

Final Answer

λ=−3,2

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