Properties of a matrix product - if AB=C and XY=C and BA=D, does YX=D?

Question

I'm trying to figure out whether a property for a matrix product exist.

Let's say we have 2 sets of matrices: $A$, $B$ and $X$, $Y$ ($A \neq X$) and let's say that their product is the same $AB=XY=C$. If we rearrange the factors, are the new products $BA$ and $YX$ equal?

If there is no such property, then are there any rules when this is true or when it is definitely false?

score 0 · Answer 1 · answered Sep 16 '17 at 11:09

0

No, the statement does not hold. For example, Consider $$ A = \pmatrix{1&0\\0&0}, \quad B = \pmatrix{1&1\\1&1}\\ X = Y = \pmatrix{1&1\\0&0} $$ Then we have $AB = XY$, but $BA \neq YX$.

The statement does hold, however, if $C$ is a non-zero multiple of the identity matrix.

answered Sep 16 '17 at 11:09

Ben Grossmann

225,327

Could you offer a simple reason or a reference why does the latter hold? – rist Sep 16 '17 at 11:19
Sure. The simple explanation is that if $AB = \lambda I$ ($\lambda$ is a non-zero real number), then $B = \frac 1{\lambda}A^{-1}$. You could see this as a generalization of the fact that if $AB = I$, then $BA = I$ (as explained in the link). – Ben Grossmann Sep 16 '17 at 11:31

Properties of a matrix product - if AB=C and XY=C and BA=D, does YX=D?

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