# Why do we define addition of matrices only when they have the same size

What happens if we define $$\begin{pmatrix} 1 & 2 \\ 1 & 2 \\ 1 & 2 \end{pmatrix} + \begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 3 \\ 1 & 2 & 3 \end{pmatrix} = \begin{pmatrix} 2 & 4 & 3 \\ 2 & 4 & 3 \\ 2 & 4 & 3 \end{pmatrix}$$ I think computers do this to refresh a part of the screen. But why can't we do it by matrices by filling the remaining rows/columns with zero? Is it related to the definition of vector spaces?

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You can do it if you find any use for it... –  Karolis Juodelė Apr 12 '14 at 4:16