Linear algebra, matrix in $\mathbb{C}^n$

Question

, Can you help me with the following exercise?

Show that doesn't exists matrix $A,B\in M_n(\mathbb{C})$ such that $$AB-BA=I$$

Has something to do with the annihilator? Thanks !

Every commutator $[A,B]=AB-BA$ has trace $0$. – Pedro Jun 25 '14 at 04:23 — Pedro, Jun 25 '14 at 04:23

score 3 · Answer 1 · answered Jun 25 '14 at 04:19

3

Had there been such matrices $A$ and $B$ such that $AB-BA=I_n$, then taking trace on both sides we will have $trace(AB-BA)=n$. But we already know that trace$(AB)$= trace($BA)$. So we will get $0=n$, which is not possible.

answered Jun 25 '14 at 04:19

tattwamasi amrutam

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Linear algebra, matrix in $\mathbb{C}^n$

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