Is it possible for both AB and BA to be identity matrices if m does not equal n?

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SUMMARY

It is established that if matrices A (of size mxn) and B (of size nxm) are such that m does not equal n, then both products AB and BA cannot simultaneously be identity matrices. The reasoning hinges on the properties of linear transformations, where AB = I implies that A is injective and BA = I implies that B is surjective. Since the dimensions of A and B are incompatible when m ≠ n, this leads to a contradiction, confirming that both products cannot be identity matrices.

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  • Understanding of matrix multiplication and dimensions
  • Knowledge of linear transformations and their properties
  • Familiarity with concepts of injectivity and surjectivity
  • Basic linear algebra, including identity matrices
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This discussion is beneficial for students studying linear algebra, educators teaching matrix theory, and anyone interested in the properties of linear transformations and their implications in higher mathematics.

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Homework Statement


Prove in general that if m does not equal n, then AB and BA cannot both be identity matrices, where A is mxn and B is nxm.


Homework Equations


None (that I know of at least).


The Attempt at a Solution


At first I thought it would be a good idea to define each element in A and B and write out some elements from AB and BA, and hope that I noticed a pattern where I would see something possible only if n=m. This proved very cumbersome and I could not get it to go anywhere.

Next I tried assuming that both AB and BA equaled identity matrices of appropriate dimensions, with the intention of deriving a contradiction, but I was unfortunately unavle to do so.

I appreciate any help, as I really don't know what to try next.
 
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Think about what the equations AB = I and BA = I imply, in terms of injectivity and surjectivity of the linear maps represented by A and B.
 

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