Explain in detail please! Let A be an n times n matrix. Define the trace of A by

Question

Explain in detail please!

Let A be an n times n matrix. Define the trace of A by the formula . That is, the trace of a matrix is the sum of the diagonal entries of the matrix. Recall that for n times n matrices A and B, tr(AB) = tr (BA). Prove that for two n times n matrices A and B, tr(A + B) = tr(A) + tr(B). Prove that for any n times n matrices A, tr(A) = tr(AT). Show that there do not exist n times n (n 0) matrices A and B such that AB - BA = In where In is the n times n identify matrix. (Hint, take the trace of the left-hand side and take the trace of the right-hand side. Show that they cannot be equal.)

Explanation / Answer

a) the diagonal elements at position ii of A+B are aii+bii, hence tr(A+B)=sum(aii+bii)=tr(A)+tr(B) by linearity of the sum

b) A and A^T has the same diagonal element , so same trace

c) tr(BA-AB)=tr(AB)-tr(BA) = 0 and tr(In)=n

so the equality cannot hold since different trace.

Related Questions

Navigate

• Browse (All)
• Browse E
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.