Consider the set \begin{equation*} M=\{A^3+B^3|A,B\in\mathcal{M}_n(\mathbb{C})\} \end{equation*} for $n\geq 1$. Prove that $(M,\cdot\;)$ is a monoid.
This is a problem I found in a section of "extra problems" in the romanian magazine Gazeta Matematica, given as a high school problem.
As mentioned in the comment, we need to show that for any complex matrices $A, B, C, D$, one needs to write
$$(A^3 + B^3)(C^3+D^3)$$
as $E^3 + F^3$ for some complex matrices $E, F$. Even in the case $n=1$, I can't algebraically find $e, f\in \mathbb C$ so that $$(a^3+ b^3)(c^3+d^3) = e^3+f^3$$ for any given $a, b, c, d\in \mathbb C$.
Let $X$ be a complex square matrix. Then $X-tI$ is invertible for some $t>0$. As every invertible complex matrix has a matrix logarithm, $X=A^3+B^3$, where $A=\exp(\frac13\log(X-tI))$ and $B=t^{1/3}I$. Hence $M=\mathcal M_n(\mathbb C)$. That is, matrix multiplication is closed in $M$. Associativity of matrix multiplication and existence of identity element should be clear.