Find the Eigenvalues [[2,1],[3,4]]

Problem

[[2,1],[3,4]]

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(2−λ)=0

2. Calculate the determinant of the 2×2 matrix using the formula a*d−b*c

(2−λ)*(4−λ)−(1)*(3)=0

3. Expand the expression by multiplying the binomials and combining like terms.

8−2*λ−4*λ+λ2−3=0

λ2−6*λ+5=0

4. Factor the quadratic equation to find the roots for λ

(λ−5)*(λ−1)=0

5. Solve for the eigenvalues by setting each factor equal to zero.

(λ_1)=5

(λ_2)=1

Final Answer

λ=5,1

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