# eigenvalue

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(a) Prove that if A and B are both invertible n x n matrices, then AB and BA have the same eigenvalues.

(b) Given that A is an n x n matrix with eigenvalues λi, i = 1, . . . , n, prove that:

n n

(i) ∑λi = tr (A) and (ii) ∏ λi = │A│

i=1 i=1

https://brainmass.com/economics/principles-of-mathematical-economics/matrices-eigenvalues-examples-11476

#### Solution Preview

See attached

(a) Prove that if A and B are both invertible n x n matrices, then AB and BA have the same eigenvalues.

One definition of an eigenvalue of an nxn matrix M, is that it is the root of this "characteristic" polynomial PM(λ):

PM (λ)= det(λIn-M).

Obviously, if AB and BA have the same characteristic polynomial, then the roots are the same, so the eigenvalues are the ...

#### Solution Summary

The expert examines an eigenvalue matrix.