Find the Eigenvalues [[9,-4],[12,-5]]

Problem

[[9,−4],[12,−5]]

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 for the given 2×2 matrix.

|[9−λ,−4],[12,−5−λ]|=0

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

(9−λ)*(−5−λ)−(−4)*(12)=0

4. Simplify the quadratic expression by performing the multiplication and combining like terms.

−45−9*λ+5*λ+λ2+48=0

λ2−4*λ+3=0

5. Factor the quadratic equation to find the roots.

(λ−3)*(λ−1)=0

6. Solve for the eigenvalues by setting each factor to zero.

(λ_1)=3

(λ_2)=1

Final Answer

λ=3,1

Want more problems? Check here!