The trace, written tr (A) of an nxn matrix A is the sum of

thediagonal elements. It can be shown that, if A and B are

nxnmatrices then tr (AB) = tr (BA).

Prove that if A is similar to B then tr (A) = tr (B).

Step-by-step explanation:

If A and B are nxn matrices then tr (AB) = tr (BA) ... Hypothesis (1)

Prove:

A is similar to B

⇒There exists an invertible n-by-n matrix P such that B=P^{-1}AP.

⇒ tr[B]=tr[P^{-1}AP]

⇒tr[B]=tr[ (P^{-1}A) P] Matrix multiplication has the associative property

⇒tr[B]=tr[P (P^{-1}A) ] Using hypothesis (1)

⇒tr[B]=tr[ (PP^{-1}) A] Matrix multiplication has the associative property

⇒tr[B]=tr[IA] I is the identity matrix

⇒tr[B]=tr[A]