# Addition of 2 matrices with different dimensions

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Christos Tsallis on 12 Mar 2021
Commented: Christos Tsallis on 12 Mar 2021
Hello,
If we want to add 2 matrices in maths, their dimensions must be the same.
I was wondering how MATLAB can add x=[1;5;9] which has dimensions of 3x1 with y=[9 2 8] which has dimensions 1x3 and the result is x+y=[10 3 9;14 7 13;18 11 17] which has dimensions 3x3.

Allen on 12 Mar 2021
In this instance your are describing, MATLAB is assuming element-wise addition between the rows and columns. Where
Where, x =
[x1
x2
x3];
and y = [y1 y2 y3];
The result of x+y then becomes:
[x1+y1 x1+y2 x1+y3
x2+y1 x2+y2 x2+y3
x3+y1 x3+y2 x3+y3]
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Christos Tsallis on 12 Mar 2021
Thanks a lot, but is this mathematically correct?

Steven Lord on 12 Mar 2021
If we want to add 2 matrices in maths, their dimensions must be the same.
I believe most texts accept a slight generalization of that, to allow adding scalars to matrices that are not 1-by-1. Scalar expansion has been part of MATLAB for longer than I've been at MathWorks (nearly 20 years.) It's probably in Cleve's original Fortran MATLAB.
A = magic(3)
A = 3×3
8 1 6 3 5 7 4 9 2
B = A + 1 % If we applied the idea you stated this should be A + ones(3)
B = 3×3
9 2 7 4 6 8 5 10 3
Implicit expansion is a generalization of that behavior. It avoids having to repmat the vectors to a common size we can compute, thus saving memory. After all, if A in my example above took up 1 GB of space (which would mean B would also take 1 GB of space) do you really want to have to allocate a temporary 1 GB matrix all of whose elements are 1?
Thanks a lot, but is this mathematically correct?
I won't tell the Mathematics Police if you don't.
Christos Tsallis on 12 Mar 2021