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?
 A: Supose $X=(x_0\;x_1\;\dots\;x_n)$, were $x_k$ are the columns of the matrix, then
$$ AX=A(x_0\;x_1\;\dots\;x_n)=B$$ 
but that is equivalent to 
$$ Ax_0=b_0$$
$$ Ax_1=b_1$$
$$\vdots$$
$$ Ax_n=b_n$$ 
where $b_k$ is the $k$-th column of the $B$ matrix. So we get a new system of equations:
$$A'\;x'=b'$$
EDIT: and A' is
$$ A'=\begin{pmatrix} A \\ & A \\ & & \ddots \\& & & A \end{pmatrix}$$
A: 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,
$$\begin{bmatrix} \mathrm A & & & \\ & \mathrm A & & \\ & & \ddots & \\ &  & & \mathrm A\end{bmatrix} \begin{bmatrix} \mathrm x_1\\ \mathrm x_2\\ \vdots\\ \mathrm x_p\end{bmatrix} = \begin{bmatrix} \mathrm b_1\\ \mathrm b_2\\ \vdots\\ \mathrm b_p\end{bmatrix}$$
A: It seems a not particularly attractive idea. Consider the system 
$$\underbrace{A}_{m \times n} \underbrace{\boldsymbol{x}}_{n \times n} = \underbrace{\boldsymbol{b}}_{m \times 1}. $$
If we were to decide to use a matrix $X: n \times m$ we would get
$$\underbrace{A}_{m \times n} \underbrace{X}_{n \times m} = \underbrace{B}_{m \times m},$$ but can we represent this system also using a regular system of equations.
