Find the Eigenvalues [[23,-5],[65,-7]]

Problem

[[23,−5],[65,−7]]

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(23−λ)=0

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

(23−λ)*(−7−λ)−(−5)*(65)=0

3. Expand the expression using the FOIL method and simplify the constant terms.

−161−23*λ+7*λ+λ2+325=0

4. Combine like terms to form a quadratic equation in standard form.

λ2−16*λ+164=0

5. Apply the quadratic formula λ=(−b±√(,b2−4*a*c))/(2*a) where a=1 b=−16 and c=164

λ=(16±√(,(−16)2−4*(1)*(164)))/(2*(1))

6. Simplify the discriminant and solve for the complex roots.

λ=(16±√(,256−656))/2

λ=(16±√(,−400))/2

λ=(16±20*i)/2

λ=8±10*i

Final Answer

λ=8+10*i,8−10*i

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