When identity matrix can be expressible as $AB-BA$?

Asked Sep 28 '19 at 06:21

Active Sep 28 '19 at 06:46

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This is very basic question; but I am actually considering another side which usually not considered.

If $A$ and $B$ are square matrices of size $\ge 2$, say over reals, or complex or rationals, then $AB-BA$ can never be identity; we consider trace of $AB-BA$ and that of identity.

But, now assume that we have a field $F$ of positive characteristic $p$; consider matrices over $F$.

Q. When it is possible to express identity matrix $I$ as $AB-BA$ for suitable square matrices $A$ and $B$?

For this, of course, the dimension/size of matrix should be divisible by $p$.

edited Sep 28 '19 at 06:46

Travis Willse

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asked Sep 28 '19 at 06:21

Beginner

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See this question. – user125932 Sep 28 '19 at 06:37
See especially this answer: https://math.stackexchange.com/a/99180/155629 – Travis Willse Sep 28 '19 at 06:40
@Travis: yes, I just saw (and found much clear answer; upvoted). – Beginner Sep 28 '19 at 06:41

When identity matrix can be expressible as $AB-BA$?

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