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

Problem

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

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 for the 2×2 matrix.

|[2−λ,1],[3,2−λ]|=0

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

(2−λ)*(2−λ)−(1)*(3)=0

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

λ2−4*λ+4−3=0

λ2−4*λ+1=0

5. Apply the quadratic formula λ=(−b±√(,b2−4*a*c))/(2*a) to solve for λ

λ=(4±√(,(−4)2−4*(1)*(1)))/(2*(1))

6. Simplify the radical and the fraction to find the two eigenvalues.

λ=(4±√(,16−4))/2

λ=(4±√(,12))/2

λ=(4±2√(,3))/2

λ=2±√(,3)

Final Answer

λ=2+√(,3),2−√(,3)

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