Question

Let T be the set of all matrices of the form AB-BA where A and B are nxn matrices. Show that span T is not Mnn. (Hint: use properties of trace)

Answer 1

Given: T is the set of all matrices with the form of AB-BA.
A and B are nxn matrices.

Proof: We need to prove that span T isn't Mnn. However, we need to obtain information from section 1.3 exercise 43.

From Section 1.3 Exercise 43

Tr(AB)=Tr(BA)
Let AB = c
Tr(AB) = Tr(C)
C=[Cij]

TR(AB)=TR(BA)
TR(AB)-TR(BA)=0
Span T is a subset of the set S of all nxn matrices with trace = 0. However, S is a proper subset of Mnn.