If $A$ and $B$ are real matrices and $X,Y$ are are non-singular square matrices with real entries such that $XA=BY$ then which of the following is true?
$3.$If $X$ and $A$ commute then $B$ is square.
$4.$If $Y$ and $A$ commute then $B$ is square.
$5.$If $A$ is non-singular then $B$ is also non-singular.
My try: Here, dimension means order of matrix. So I suppose that $$\dim(X)=m\times m, \quad\dim(Y)=n\times n, \quad\dim(A)=m\times a, \quad\dim(B)=n\times b$$
Then I try to think about the options but it becomes difficult for me. Last option is obviously true and I think first two options are also true. What you think? What should be the answer?