How Can XY = 1 and YZ = 1 Prove X = Z?

[ConeOfIce]

Homework Statement

Suppose one has n×n square matrices X, Y and Z such that
XY = 1and Y Z = 1. Show that it follows that X = Z.

The Attempt at a Solution

Now I know if the equatoins had been XY and ZY I would do this:

XY=ZY -> XY-ZY=0 -> Y(X-Z)=0 -> X-Z=0 -> X=Z

I was wondering if this holds when the Y is on opposite sides of the other matrices?

Thanks in advanced!

 

[Saladsamurai]

Homework Statement

Suppose one has n×n square matrices X, Y and Z such that
XY = 1and Y Z = 1. Show that it follows that X = Z.

The Attempt at a Solution

Now I know if the equatoins had been XY and ZY I would do this:

XY=ZY -> XY-ZY=0 -> Y(X-Z)=0 -> X-Z=0 -> X=Z

I was wondering if this holds when the Y is on opposite sides of the other matrices?

Thanks in advanced!

I am sure you would have better luck in the math forums.

Casey

 

[D H]

You are throwing away a lot of information in going from XY=1 and ZY=1 to XY=ZY. If Y is singular, there are infinitely many distinct matrices X and Z for which XY=ZY. With matrices and vectors, the step (X-Z)Y = 0 to X-Z=0 is in general invalid.

Hint: What is XYZ?