Find the Eigenvalues

Problem

[[19,25,30,5],[−23,−30,−35,−5],[7,9,10,1],[−3,−4,−5,−1]]

Solution

1. Set up the characteristic equation by subtracting λ from the diagonal elements of matrix A and setting the determinant of A−λ*I to zero.

det(A−λ*I)=|[19−λ,25,30,5],[−23,−30−λ,−35,−5],[7,9,10−λ,1],[−3,−4,−5,−1−λ]|=0

2. Simplify the determinant using row operations. Add row 2 to row 1 to create zeros and simplify the expressions.

(R_1)←(R_1)+(R_2)

|[−4−λ,−5−λ,−5,0],[−23,−30−λ,−35,−5],[7,9,10−λ,1],[−3,−4,−5,−1−λ]|=0

3. Perform further row operations to simplify the fourth column. Add 5 times row 3 to row 2.

(R_2)←(R_2)+5*(R_3)

|[−4−λ,−5−λ,−5,0],[12,15−λ,15−5*λ,0],[7,9,10−λ,1],[−3,−4,−5,−1−λ]|=0

4. Expand the determinant along the fourth column and solve the resulting polynomial equation.

λ4+2*λ3+λ2=0

5. Factor the characteristic polynomial to find the roots.

λ2*(λ2+2*λ+1)=0

λ2*(λ+1)2=0

6. Identify the eigenvalues from the factored form.

(λ_1)=0,(λ_2)=0,(λ_3)=−1,(λ_4)=−1

Final Answer

λ=0,0,−1,−1

Want more problems? Check here!