Nonexistence of Complex Matrices whose Commutator equals Identity

Theorem
Let $d \in \N_{> 0}$ be a positive natural number.

Let $\mathbf A, \mathbf B \in \C^{d \times d}$ be complex matrices.

Let $\mathbf I$ be the $d \times d$ identity matrix.

Then there is no $\mathbf A$, $\mathbf B$ such that $\mathbf A \mathbf B - \mathbf B \mathbf A = \mathbf I$.

Proof
We have that complex numbers form a commutative ring.

there are $\mathbf A$, $\mathbf B$ such that $\mathbf A \mathbf B - \mathbf B \mathbf A = \mathbf I$.

Then:

This is a contradiction.