The sum and the product of two quadratic matrices

Let $p$ and $q$ be polynomials with degree $2$ over an arbitrary field $\mathbb{F}$. In the first part of this article, we characterize the matrices that can be decomposed as $A+B$ for some pair $(A,B)$ of square matrices such that $p(A)=0$ and $q(B)=0$. The case when both polynomials $p$ and $q$ are split was already known. In the first half of this article, we complete the study by tackling the case when at least one of the polynomials $p$ and $q$ is irreducible over $\mathbb{F}$. In the second half of the article, we use a similar method to characterize, under the assumption that $p(0)q(0)\mathrm{\ne }0$, the matrices that can be decomposed as $AB$ for some pair $(A,B)$ of square matrices such that $p(A)=0$ and $q(B)=0$.