Matrix Multiplication: Can You Multiply a Matrix in the Middle?

[EvLer]

Ok, maybe a dumb question, but anyway:
if I have some matrices equation like this:

AB = BC

if I want to multiply both sides by some matrix Z can I stick it in the middle?

AZB = BZC

I know I can do these two:

ABZ = BCZ
ZAB = ZBC

but can I do the middle too?

Thanks!

 

[Hurkyl]

Have you tried some examples to see if it works?

 

[EvLer]

Have you tried some examples to see if it works?

no
but the proof worked out nice, so I am wondering if I am actually not doing it right.
Good point.
I just rely on the fact that people here know everything and know it off top of their heads, so it would save me trouble
But usually, I make an effort before I ask here. I am just pressed for time: reviewing before finals.