Inhomogeneous multiplicative Diophantine approximation on matrix approximation

In this paper, we establish a coherent theory for inhomogeneous multiplicative Diophantine approximation on matrix approximation. More specifically, for any $n, m\in\mathbb{N}$ and $\textbf{y}\in [0, 1]^n$, let $\psi: \mathbb{N}\rightarrow [0, \infty)$ be a positive non-increasing function, and $\alpha_1, \alpha_2, \dots, \alpha_n$ be positive reals with $A(n)=\alpha_1+\cdots+\alpha_n$. A dichotomy law of the Hausdorff measure for the following set \begin{align*} \mathcal{M}_{n, m}^{\textbf{y}}(\psi; \alpha_1, \dots, \alpha_n):=\Bigg\{\textbf{x}\in[0, 1]^{nm}: \prod\limits_{i=1}^n\|q_1x_{i1}&+\cdots+q_mx_{im}-y_i\|^{\alpha_i}\\&＜\psi(|\textbf{q}|)^{A(n)} \text{ for i.m.}\, \textbf{q}\in \mathbb{Z}^m \Bigg\} \end{align*} is obtained, which depends on the convergence or divergence of a certain series.