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Here , the actual question is to find any matrices $A$ and $B$ such that $AB-BA=I$ relation holds.

but actually, I dont think that we could not find any such matrices. As, the diagonal elements of the matrices $A$ and $B$ would be same. so, after the subtraction, the diagonal elements of the matrics $AB-BA$ would be all $0$. so, the trace of the matrics would be actually zero, where in the right hand side, the trace of the matrics $I$ would be $n$.

so, we could not find any such matrices $A$ and $B$.Is my approach is correct?

ROBINSON
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  • Why would the diagonal elements of $A$ and $B$ have to be the same? It's true that the trace of $AB-BA$ always equals $0$, but not for this reason. – Greg Martin Apr 11 '14 at 07:07
  • yeah, I got it, thanks for giving the link of original question(from artin).@NajibIdrissi – ROBINSON Apr 11 '14 at 07:08
  • Did you take the characteristic of the ground ring into account? The $n\times n$ identity matrix has trace $0$ in characteristic $n$. – Oliver Braun Apr 11 '14 at 07:53

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