Why is the $O$ (zero) matrix important? In reading my linear algebra book I found it quite interesting that they made the following comment:

One important property of addition of real numbers is that the number $0$ is the additive identity...

It then went on to relate this property to matrices.
But all the proofs relating to it seem so obvious that I don't understand why they precursor it with important.
$A+O_{mn}=A$ seems to0 painfully obvious to me that I feel like I'm missing the bigger picture in the application of this property.
Am I missing something or are there certain applications of the zero matrix that prove it's importance?
 A: One way of seeing the importance of an additive identity is because it allows basic algebraic manipulations to go on. Suppose you have a simple matrix equation: $A+B=C$, and you want to solve for $A$. The steps, in detail, are this:
First, you note that there exists a matrix $-B$, which is the additive inverse of the matrix $B$. If you add this to both sides, you obtain $A+B+(-B)=C+(-B)$. Next, we use the property of additive inverses to simplify $B+(-B)$ into the zero matrix. This gives us $A+O=C+(-B)$. Finally, because $O$ is the additive identity, we can replace $A+O$ with just $A$, yielding $A=C+(-B)$, or in the more usual way of writing it, $A=C-B$.
If you were working in some system where you didn't have an additive identity, then you couldn't perform operations as simple as canceling something with its opposite. We do it so naturally and frequently that we don't always think about it, but an additive identity does a lot of work for us, algebraically.
A: Important things don't have to be complicated.
To me, the existence of the zero matrix $O_{mn}$ allows us to define the collection of $m\times n$ matrices $\mathcal M_{m\times n}$ as an additive group. Moreover, the existence of the zero matrix $O_{nn}$ allows us to define the collection of $n\times n$ matrices $\mathcal M_{n\times n}$ as a ring
A: The fact that we have a zero matrix is important because we can study the structure of matrices as a type of algebraic structure that has unit. And since the addition of matrices is associative and we have negative matrices (for matrice over a ring) we can apply results of group theory to matrices. The fact that the zero matrix exist and it follows the properties we need has to be proven, and the fact that the proof is elemntary does not make the result less important. It is of tremendous importance otherwise  if one axiom of group theory isnt satisfied all that would be useless to us.
