Consider the following two statements with respect to the matrices Am×n , Bn×m , Cn×n and Dn×n
Statement 1: tr(AB) = tr(BA) Statement 2: tr(CD) = tr(DC)
where tr() represents the trace of a matrix. Which one of the following holds?
(A)
Statement 1 is correct and Statement 2 is wrong.
(B)
Statement 1 is wrong and Statement 2 is correct.
(C)
Both Statement 1 and Statement 2 are correct.
(D)
Both Statement 1 and Statement 2 are wrong.
Answer: (C)
Explanation:
The eigenvalues (counting multiplicity) of AB are the same as those of BA
By theorem cited in the second edition of ”Matrix Analysis” by Horn and Jhonson says, If A is an m by n matrix and B is an n by m matrix with n>m then the characteristic polynomial PBA of BA is related to the characteristic polynomial PBA of AB by
PBA(t) - tn m PBA
Since Eigenvalues are the same, thus the sum of Eigen Values(i.e., the Traces) is also the same
So, option C is the correct answer.
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