uniqueness of neutral element for matrix addition confusion I'm reading Basic Linear Algebra 2e (T.S.Blyth and E.F.Robertson) and have come across the following theorem:

Theorem
There is a unique $m \times n$ matrix $M$ such that, for every $m \times n$ matrix $A$ one has $A + M = A$.
Proof
Consider the matrix $M = [m_{ij}]_{m  \times n}$ all of whose entries are $0$; i.e. $m_{ij} = 0$ for all $i,j$.
For every matrix $A = [a_{ij}]_{m \times n}$ we have:
$A + M = [a_{ij} + m_{ij}]_{m  \times n} = [a_{ij} + 0]_{m  \times n} =[a_{ij}] = A$
To establish the uniqueness of this matrix M, suppose that $B = [b_{ij}]_{m \times n}$ is also such that $A + B = A$ for every $m \times n$ matrix $A$.  Then in particular we have $M + B = M$.

Question: I understand everything up to but not including the last sentence.  How do we get to $M + B = M$?  Please dumb down the answer as much as possible - my math is very rusty!

This uses a technique very common in uniqueness proofs. It goes like this: 


*

*Assume $X$ and $Y$ both have the properties that we want

*Show that actually, $X = Y$, so $X$ was unique.
In this case, they are using the fact that that $A + B = A$ for any matrix $A$ to say that $M + B = M$. This is allowed because $M$ falls under the umbrella of "any matrix."
Does that make sense? feel free to ask questions. 
