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

Problem

[[3,4],[−2,−4]]

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 expression for the 2×2 matrix.

|[3−λ,4],[−2,−4−λ]|=0

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

(3−λ)*(−4−λ)−(4)*(−2)=0

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

−12−3*λ+4*λ+λ2+8=0

λ2+λ−4=0

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

λ=(−1±√(,1−4*(1)*(−4)))/(2*(1))

λ=(−1±√(,1+16))/2

λ=(−1±√(,17))/2

Final Answer

λ=(−1+√(,17))/2,(−1−√(,17))/2

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