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$A=\begin{pmatrix} 2 & 5 & -3\\ 0 &-1 & 6 \end{pmatrix}$

$B=\begin{pmatrix} 3 & 1\\ 4 & -2\\ 0 & 6 \end{pmatrix}$, $A,B \in \mathbb{R}$

Is $A+B$ possible?

I have tried it on paper but I cannot decide how it could be done correctly : /

So now I'm wondering if this is possible defined at all?

By the way, would it be possible to do $A+1$ when $1 \in \mathbb{R}$

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    $A,B\in\mathbb{R}$? ? – 2017-01-08
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    @Jack both are real matrices – 2017-01-08
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    No. One would never ever use a notation in that way. – 2017-01-08
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    There is a good question at the heart of this. But @Jack has hit upon your main misconception already. You could meaningfully compute $A+B^{T} $ but why would you want to? – 2017-01-08

3 Answers 3

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No, it is not. The rule for matrix addition is element-wise. You can multiply them though!

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    Thanks!!!! Would A+1 work? Because only one element and we can imagine the other as zero so same size? – 2017-01-08
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    Not unless $A$ happened to be a 1 by 1 matrix. If you mean a matrix with 1 in the first entry and zeros everywhere else, you should just write that so as to avoid confusing people. – 2017-01-08
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You're confusing things: $A,B$ are not $\in\Bbb{R}$. The numbers in $A,B$ are $\in \Bbb{R}$. The notation you are employing to say that those are real matrices does not exist in - at least - decent literature on the subject.

In the standard way the summation of matrices is defined ,which is based on sum of linear transformations, you can't sum that two matrices. Also, If $1$ is a number in $\Bbb{R}$, you can't sum it with a matrix unless you define it in some meaningful way.

I recommend you to review matrix operations. You can use anything on the internet, perhaps Khan academy. If you need a book, take Lang's: Introduction to Linear Algebra.

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I think Ken Kuttler in his book A First Course in Linear Algebra (2014) puts it quite well. I'll refer to Definition 2.5 on page 61

Let $A = [a_{ij}]$ and $B = [b{ij} ]$ be two $m × n$ matrices. Then $A + B = C$ where $C$ is the $m × n$ matrix $C = [c_{ij} ]$ defined by $$c_{ij} = a_{ij} + b_{ij}$$

When rows and columns are of the same size, we are able to preform Matrix addtion. Eitherwise it is not possible.