Conference rusure::math

263.0. "Finding BA given AB" by HARE::STAN () Wed Apr 24 1985 14:38

I was recently shown a 3 X 3 matrix, C, (I forget the values of the entries)
with the following property:

	I was told that C was the product of a 3 X 2 matrix, A, and
	a 2 X 3 matrix, B.  That is, C=AB.  I was further told that
	this was enough information for me to find BA.
	[BA is a 2 X 2 matrix.]

Sure enough, I followed through the proof, and it turned out that BA
was uniquely determined, knowing only the value of AB.  Note that is
not true for arbitrary C; but it \was/ true for the particular C that
I was given.  And by the way, C was not trivial, that is, it was
not a scalar multiple of the identity matrix.

Anyhow, the question I now have is this:  Characterize those 3 X 3
matrices, C, that have this property (that knowing C=AB, uniquely
determines the value of BA).

Perhaps an easier question might be of some use:  Suppose I have a non-trivial
2 X 2 matrix, C.  Is it possible that knowing C=AB where A and B are
each (unknown) 2 X 2 matrices can uniquely determine the value of BA?