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

Problem

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

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

|[4−λ,3],[2,3−λ]|=0

2. Calculate the determinant by multiplying the diagonal elements and subtracting the product of the off-diagonal elements.

(4−λ)*(3−λ)−(3)*(2)=0

3. Expand the expression to form a quadratic equation in terms of λ

12−4*λ−3*λ+λ2−6=0

λ2−7*λ+6=0

4. Factor the quadratic equation to find the roots, which are the eigenvalues.

(λ−6)*(λ−1)=0

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

(λ_1)=6

(λ_2)=1

Final Answer

λ=6,1

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