Find the Eigenvalues [[3,0],[1,-2]]

Problem

[[3,0],[1,−2]]

Solution

1. Set up the characteristic equation by subtracting λ from the diagonal elements of the matrix and setting the determinant of the resulting matrix to zero.

det(A−λ*I)=0

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

det(3−λ)=0

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

(3−λ)*(−2−λ)−(0)*(1)=0

4. Simplify the equation to find the characteristic polynomial.

(3−λ)*(−2−λ)=0

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

3−λ=0⇒(λ_1)=3

−2−λ=0⇒(λ_2)=−2

Final Answer

λ=3,−2

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