Answer
There do not exist matrices $A,B$ such that $AB-BA=I$
Work Step by Step
Since $A$ and $B$ are square matrices of order $2$,
$\therefore Tr(AB)=tr(BA)$
If $AB-BA=I$ (where $I$ represents the identity matrix) is true, then
$Tr(AB-BA)=Tr(I)\\
Tr(AB)-Tr(BA)=Tr(I)\\
$
but the $LHS =0$ and $RHS=1$.
Hence, there do not exist matrices $A,B$ such that $AB-BA=I$