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]