Show that if A is an [m x n] matrix and A(BA) is defined, then B is an [n x m] matrix.
If A has order $\displaystyle m\times n$ look at BA first of all
For be BA to be defined B must have order $\displaystyle k\times m$ in turn BA must have order $\displaystyle k\times n$
Now for A(BA) to be defined $\displaystyle k=n$ therefore B has order $\displaystyle n\times m$