Question

Prove that (AB)' is not same as (BA)'.​

Answer 1

Answer:

Let  A  be  m×n. Since  AB  and  BA  both exist, hence  B  must be  n×m. Thus,  AB  is  m×m  and  BA  is  n×n. So far, all we have established is both  AB  and  BA  are square matrices. If  m≠n  then  AB  and  BA  have different sizes. They can’t be equal. If  m=n  then  AB  and  BA  can be compared but in general  AB≠BA  since matrix multiplication is non-commutative in general. In the special case that  A  and  B  commute with each other, you will have  AB=BA.  

Two matrices are said to commute if  AB=BA  or equivalently, their commutator  [A,B]=AB−BA  is 0. There is no straightforward answer to prove if the matrices commute. However, there are some special examples which may be relevant for you. We are assuming here that both  A and  B are square matrices with same dimensions. Identity matrix commutes with every matrix . A diagonal matrix commutes with every matrix. If A and  B are both symmetric and their product AB is also symmetric then  AB=BA.

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