Example for quasi-commuting matrices

Question

I can not find example of 2 matrices that quasi commute; $[A,B]=AB−BA = c I$, where $c$ is a scalar not equal to $0$ and $I$ is the identity matrix. As far as I know there is no 2x2 matrices that satisfy the quasi commutative property. Any help for the 3x3 matrices?

Answer 1

$$\DeclareMathOperator{\Tr}{Tr}$$ As pointed out in the comments, there are no finite-dimensional examples over a field of characteristic $0$, since $$0=\Tr\,[A,B]=c\Tr I=cn$$ which implies $c=0$ if $n$ is a nonzerodivisor.

For an infinite-dimensional example, consider the operators $$\begin{align} D &: \mathbb{Q}[x]\to\mathbb{Q}[x] & Dp(x)&=p'(x)\\ X &: \mathbb{Q}[x]\to\mathbb{Q}[x] & Xp(x)&=x\cdot p(x)\text{.} \end{align}$$ on the ring $\mathbb{Q}[x]$ of univariate polynomials with coefficients from $\mathbb{Q}$. Since $\mathbb{Q}[x]\cong \mathbb{Q}^{\oplus \mathbb{N}}$ qua $\mathbb{Q}$-vector spaces (pick the monomial basis), with respect to this isomorphism $D$ and $X$ admit infinite-dimensional matrix representations. Furthermore, $$DX-XD=I\text{.}$$