SOLUTION: Prove that if λ is an eigencalue of an invertible matrix A and x is a corresponding eigenvector, then 1/λ is an eigenvalue of A inverese (A(-1)) , and x is a correspondin

Algebra ->  Algebra  -> College  -> Linear Algebra -> SOLUTION: Prove that if λ is an eigencalue of an invertible matrix A and x is a corresponding eigenvector, then 1/λ is an eigenvalue of A inverese (A(-1)) , and x is a correspondin      Log On


   

Question 35533: Prove that if λ is an eigencalue of an invertible matrix A and x is a corresponding eigenvector, then 1/λ is an eigenvalue of A inverese (A(-1)) , and x is a corresponding eigenvector
Answer by narayaba(40) About Me  (Show Source):
You can put this solution on YOUR website!
Given that λ is an eigenvalue of the invertibe matrix with x as its eigen vector.
This means Ax = λx such that x is non-zero
Ax = λx
lets multiply both side of the above equation by the inverse of A( A^-1) from the left. This is possibe since the inverse of A exits according to the problem definition.
(A^-1)*A*x = (A^-1)*λx
since (A^-1)*A = I(identity matrix)
(A^-1)*λx = x
muliply by 1/λ both sides
(A^-1)*x = 1/λx
this shows that 1/λ is eigenvalue of matrix A^-1 with eigen vector x