Question

The trace of a square matrix A is the sum of the diagonal entries in A and is denoted by tr A. It can be verified that ​tr(FG)equals​tr(GF) for any two n times n matrices F and G. Show that if A and B are​ similar, then tr Upper A equals tr Upper B.

Answer 1

Answer with Step-by-step explanation:

We are given that tr(FG)=tr(GF) for any two matrix of order $n\times n$

We have to show that if A and B are similar then

tr upper A=tr upper B

Trace of a square matrix A is the sum of diagonal entries in A and denoted by tr A

We are given that A and B are similar matrix  then there exist a inverse matrix P such that

Then $B=P^{-1}AP$

Let $G=P^{-1}$ and F=AP

Then$FG= APP^{-1}$=A

GF=$P^{-1}AP=B$

We are given that tr(FG)=tr(GF)

Therefore, tr upper A=trB

Hence, proved