The vector space $M_{mn}$ is the set of all $m\times n$ matrices with entries from the set of complex numbers.
Just as we made, and used, a careful definition of equality for column vectors, so too, we have precise definitions for matrices.
The $m\times n$ matrices $A$ and $B$ are equal, written $A=B$ provided $\matrixentry{A}{ij}=\matrixentry{B}{ij}$ for all $1\leq i\leq m$, $1\leq j\leq n$.
So equality of matrices translates to the equality of complex numbers, on an entry-by-entry basis. Notice that we now have yet another definition that uses the symbol “=” for shorthand. Whenever a theorem has a conclusion saying two matrices are equal (think about your objects), we will consider appealing to this definition as a way of formulating the top-level structure of the proof.
We will now define two operations on the set $M_{mn}$. Again, we will overload a symbol (`+') and a convention (juxtaposition for scalar multiplication).
Given the $m\times n$ matrices $A$ and $B$, define the sum of $A$ and $B$ as an $m\times n$ matrix, written $A+B$, according to \begin{align*} \matrixentry{A+B}{ij}&=\matrixentry{A}{ij}+\matrixentry{B}{ij} &&1\leq i\leq m,\,1\leq j\leq n \end{align*}
So matrix addition takes two matrices of the same size and combines them (in a natural way!) to create a new matrix of the same size. Perhaps this is the “obvious” thing to do, but it does not relieve us from the obligation to state it carefully.
Our second operation takes two objects of different types, specifically a number and a matrix, and combines them to create another matrix. As with vectors, in this context we call a number a scalar in order to emphasize that it is not a matrix.
Given the $m\times n$ matrix $A$ and the scalar $\alpha\in\complex{\null}$, the scalar multiple of $A$ is an $m\times n$ matrix, written $\alpha A$ and defined according to \begin{align*} \matrixentry{\alpha A}{ij}&=\alpha\matrixentry{A}{ij}&& \quad 1\leq i\leq m,\,1\leq j\leq n \end{align*}
Notice again that we have yet another kind of multiplication, and it is again written putting two symbols side-by-side. Computationally, scalar matrix multiplication is very easy.