Question

If square matrices $A$ and $B$ are such that $A{A}^{\prime }=A^{\prime }A,B{B}^{\prime }=B^{\prime }B,A{B}^{\prime }=B^{\prime }A$,
then is the statement $AB(AB{)}^{\prime }=(AB{)}^{\prime }AB$ is
where $A^{\prime }$ is transpose of $A$
If true enter 1 else enter 0

Answer 1

Given $A{A}^{\prime }=A^{\prime }A,B{B}^{\prime }=B^{\prime }B,A{B}^{\prime }=B^{\prime }A$
$(A{B}^{\prime }{)}^{\prime }=(B^{\prime }A{)}^{\prime }$
$\Rightarrow B{A}^{\prime }=A^{\prime }B$ ------(1)
Now,
$AB(AB{)}^{\prime }=AB{B}^{\prime }{A}^{\prime }=A{B}^{\prime }B{A}^{\prime }=B^{\prime }A{A}^{\prime }B$ from (1)
$=B^{\prime }{A}^{\prime }AB=(AB{)}^{\prime }AB$
$\therefore AB(AB{)}^{\prime }=(AB{)}^{\prime }AB$ is true.

Hence, the answer is $1$.