# why can’t you add matrices of different sizes?

Why can’t matrices of different shapes be added together? For example if you have these 3 equations:

x+y=1
x+y=1

and

x+y+z=3

the corresponding coefficient matrices are

âژ،1 1âژ¤
âژ£1 1âژ¦

âژ، âژ¤
|1 1 1|
âژ£ âژ¦

You can easily add the equations (first with the first one and second with the second one) to produce

2x+2y+z=4
2x+2y+z=4

, but you cant add matrices to produce

âژ،2 2 1âژ¤
âژ£2 2 1âژ¦

Why? I picked the equations randomly. I am aware of the rules, I just don’t know why are they are as they are.

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I think the primary reason is that a matrix has no sense of the intended variables behind it. You can apply an interpretation of a given matrix that allows this kind of addition, but that interpretation is specific to some goal you have and not a natural property of matrices…
– abiessu
2 days ago

1

Actually, your first matrix should be (110110)\begin{pmatrix} 1 & 1 & 0 \\ 1 & 1 & 0 \end{pmatrix}
– Ng Chung Tak
2 days ago

2

You only have two equations. Still, though, given those two equations above, what you really have is x+y+z=1+z=3⟹z=2x+y+z=1+z=3 \implies z=2 so you don’t really have 44 equations above. Anyway, the purpose of not being able to add them is that matrices live in spaces of certain dimensions, these dimensions need to be equal in order to add them, not necessary so with multiplying them, however.
– Bacon
2 days ago

1

They can’t be added because they were not “defined” so.Once Matrix addition is a “definition”, there is no questioning it. You may question the theorems evolving but not the definition.
– Bijesh K.S
2 days ago

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