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Is this something you can do?

[caij + cbij] = [caij] + [cbij]

I know you can do something like [aij + bij] = [aij] + [bij], but can you still separate out the terms like that if there's a scalar in there?

The reason I ask is 'cause just trying to understand a line in my book that says [caij + cbij] = cA + cB. I just want to make sure I know how they got there :)

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$[ca_{ij}]$ is just another matrix, right? So we could just call it, I don't know, $M = [m_{ij}]$ where $m_{ij} = ca_{ij}$. Similarly, we could define $N = [n_{ij}]$ such that $n_{ij} = cb_{ij}$. All valid so far. And as such, we can obviously write the elements of $M + N$ as $m_{ij} + n_{ij} = ca_{ij} + cb_{ij} = c(a_{ij} + b_{ij})$.

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Yes, $[c a_{ij}+ c b_{ij}]$ is understood as shorthand for: $$\begin{bmatrix}ca_{11}+cb_{11} & \cdots & ca_{1j}+cb_{1j}&\cdots & ca_{1m}+cb_{1n} \\ \vdots & \ddots & \vdots &\ddots & \vdots \\ ca_{i1}+cb_{i1} & \cdots & ca_{ij}+cb_{ij}&\cdots & ca_{im}+cb_{im}\\ \vdots & \ddots & \vdots &\ddots & \vdots \\ ca_{n1}+cb_{n1} & \cdots & ca_{nj}+cb_{nj}&\cdots & ca_{nm}+cb_{nm}\end{bmatrix}$$

Because, who would want to have to express that every time?

Sometimes we also add dimensions of the matrix after the last square bracket, $[\ldots]_{nm}$, though these are often omitted when understood by context.

So, clearly $[ca_{ij}+cb_{ij}] = c[a_{ij}]+c[b_{ij}]$, via the definitions of scalar multiplication and matrix addition.

$\ddot\smile$

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