Let A and B be n times n matrices. Recall that the trace of A , written tr(A),equal sum_{i=1}^nA_{ii} Prove that tr(AB)=tr(BA) and tr(A)=tr(A^t)

Haven 2020-12-02 Answered
Let A and B be n×n matrices. Recall that the trace of A , written tr(A),equal
i=1nAii
Prove that tr(AB)=tr(BA) and tr(A)=tr(At)
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

yagombyeR
Answered 2020-12-03 Author has 92 answers
Step 1
Consider A and B be n×n matrices and trace of matrix is defined as,
tr(A)=i=1nAii
step 2
To prove tr(AB)=tr(BA).
tr(AB)=i=1n(AB)ii
=i=1nj=1naiibjj
=j=1ni=1nbjjaii
=i=1n(BA)ii
=tr(BA)
Hence, it is proved
Step 3
To prove tr(A)=tr(At)
Let A=(aii),1in and AT=(bii) Then, bii=aii
tr(A)=i=1n(A)ii
=i=1naii
=i=1nbii
=i=1n(AT)ii
=tr(AT)
Hence, it is proved
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more