A note on products of finite-dimensional quadratic matrices

Let $F$ be a field, $n$ be a positive integer, and $q(x) = (x-\lambda_1)(x-\lambda_2)$, where $\lambda_1$ and $\lambda_2$ are two nonzero elements in $F$. Denote by $\mathbb{M}_n(F)$ the ring of all $n \times n$ matrices over $F$. A matrix $A \in \mathbb{M}_n(F)$ is called quadratic with respect to $q(x)$ if $q(A) = 0$. In this paper, we investigate the question of when a matrix in $\mathbb{M}_n(F)$ can be expressed as a product of quadratic matrices with respect to $q(x)$. First, we prove that if $F$ is a field with more than $n+1$ elements, $k \ge 0$ is an integer, and $A \in \mathbb{M}_n(F)$ has determinant $\lambda_1^{s+2n}\lambda_2^{t+2n}$, where $s, t \ge 0$ are integers such that $s + t = kn$, then $A$ can be expressed as a product of $k+4$ quadratic matrices with respect to $q(x)$. In particular, if $\lambda_1 = 1$, $\lambda_2^r = 1$ for some integer $r \geq 2$, and $A \in \mathbb{M}_n(F)$ has a determinant that is a power of $\lambda_2$, then $A$ can be expressed as a product of at most $2r$ quadratic matrices with respect to $q(x)$. As a corollary, we derive results on the factorization of matrices as products of certain special quadratic matrices.