1. ## Notation question

How are these different?

$AB=[c_{ij}]$
where $c_{ij}=\sum_{k=1}a_{ik}b_{kj}$

and

$AB=[AB]_{ij}=\sum_{k=1}[A]_{ik}[b]_{kj}$

For instance,
My book proves the distributive property of multiplication as follows (im going to leave stuff out)

$A(B+C)=a_{i1}(b_{1j}+c_{1j})+...+a_{in}(b_{in}+c_{ nj})$

Then you expand AB+AC and do some regrouping to find A(B+C)=AB+AC

But, in another text I found the same proof done with the afore mentioned notation

$[A(B+C)]_{ij}=[A]_{*i}[(B+C)]_{*j}=\sum_{k}[A]_{ik}[B+C]_{kj}...$

I don't understand how to read the latter proof. What's being done in the proof makes sense to me, but I'm just not sure how to read the notation I guess.

2. This might be of some help.

$[A]_{ij}$ represents $(i,j)$ element of the matrix.

$[A]_{i*}$ represents $i^{th}$ row of the matrix

$[A]_{*j}$ represents $j^{th}$ column of the matrix

$
[A(B+C)]_{ij}=[A]_{*i}[(B+C)]_{*j}=\sum_{k}[A]_{ik}[B+C]_{kj}...
$
it is $[A]_{i*}$ in place of $[A]_{*i}$

3. Essentially they are the same with $[A]_{ij}=a_{ij}$, and as malaygeol sort of said, $[A]_{i*}=\sum_{k}a_{ik}$.

Also, $[c_{ij}]$ is the entire matrix while $[C]_{ij}$ is just one element of it, and also (I presume) the $c$ such that $AB=[c_{ij}]$ is different from the $c$ such that $C=[c_{ij}]$, if that makes sense...

With notation problems like this I tend to just guess - if you understand what it is mean to be then you can often work out what the notation means. If it makes sense it is probably true.

4. Thank you both, everything makes sense now. You pretty much solidified what I thought. I just wasn’t 100% sure.

Thanks