Question

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).

Answer 1

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]