Is there a system of equations analogous to a matrix multiplication? I'm self studying linear algebra, and one of the basic tenets is that a system of equations can be represented by $A x = b$. A lot of properties are also based on this system, such as consistence, null space, etc. However, what if the variables are grouped in a matrix instead of a vector $x$, for example, in a $3 \times 3$ matrix? Is there any way of representing this matrix $X$ as a vector for the purposes of this multiplication and to infer if the system is over/under determined? Is it possible to represent a matrix multiplication as a system of equations?

Suppose we have the following linear matrix equation in $\mathrm X \in \mathbb R^{n \times p}$

$$\rm A X = B$$

where matrices $\mathrm A \in \mathbb R^{m \times n}$ and $\mathrm B \in \mathbb R^{m \times p}$ are given. Vectorizing) both sides, we obtain a system of $m p$ linear equations in $n p$ unknowns

$$\left( \mathrm I_p \otimes \mathrm A \right) \, \mbox{vec} (\mathrm X) = \mbox{vec} (\mathrm B)$$

or, less economically,

$$\begin{bmatrix} \mathrm A & & & \\\ & \mathrm A & & \\\ & & \ddots & \\\ & & & \mathrm A\end{bmatrix} \begin{bmatrix} \mathrm x_1\\\ \mathrm x_2\\\ \vdots\\\ \mathrm x_p\end{bmatrix} = \begin{bmatrix} \mathrm b_1\\\ \mathrm b_2\\\ \vdots\\\ \mathrm b_p\end{bmatrix}$$