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

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SUMMARY

The discussion centers on proving that if two equations, XY = 1 and YZ = 1, hold for n×n square matrices X, Y, and Z, then it follows that X = Z. The initial attempt at a solution incorrectly assumes that the manipulation of matrices with Y on opposite sides is valid. However, it is established that if Y is singular, the conclusion does not necessarily hold, as there can be infinitely many distinct matrices X and Z satisfying the equations. The key insight is to consider the product XYZ to explore the relationship between the matrices.

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ConeOfIce
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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!
 
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ConeOfIce said:

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
 
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?
 

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