Is this possible? AB- BA=I I have just started linear functionals when I faced the following problem: If $A$ and $B$ are $n \times n$ complex matrices, show $AB - BA=\Bbb{I}$ is impossible. Can someone help me?

For a matrix $A=[a_{ij}]$ of size $n\times n$, its trace $Tr(A)$ is defined by $$ Tr(A)=\sum_{i=1}^n a_{ii} $$ . You can verify it yourself that $$ Tr(AB)=Tr(BA)$$ and that $$ Tr(A+B)=Tr(A)+Tr(B) $$

Therefore if $AB-BA = \Bbb I$, then we have $$n=Tr(\Bbb I)= Tr(AB-BA)= Tr(AB)-Tr(BA) = 0 $$ which is impossible.