I have two matrices;
$$ 1) \left( \begin{matrix} 3 & -2 \\ 1 & 2 \\ 0 & -3\\ \end{matrix} \right) $$
And$$ 2) \left( \begin{matrix} -2 & 1 & 3 \\ 4 & 3 & 2 \\ \end{matrix} \right) $$
But can these two be added together, is this possible? I found from this PDF the following quote:
The sum A + B of two matrices A, B (which must have the same number of rows, and the same number of columns) is the matrix (also with the same shape) given by adding corresponding elements of A and B.
This suggests to me that I cannot add together Matrix 1 and 2, but I can however add together Matrix 2 and a third matrix with the same dimentions, for example:
$$ 3) \left( \begin{matrix} 2 & 1 & -1 \\ -1 & 0 & 3 \\ \end{matrix} \right) $$
I've come to believe that I can add up Matrix 2+3 but not Matrix 1+2 or 1+3 for instance. Am I correct in my assumptions here?
And would this differ in any way if i try to multiply the matrices?
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$\begingroup$First of all, you can't add two matrices if they don't have the same dimension. In your case,$$\left( \begin{matrix} -2 & 1 & 3 \\ 4 & 3 & 2 \\ \end{matrix} \right) + \left( \begin{matrix} 2 & 1 & -1 \\ -1 & 0 & 3 \\ \end{matrix} \right)=\left( \begin{matrix} -2+2 & 1+1 & 3+(-1) \\ 4+(-1) & 3+0 & 2+3 \\ \end{matrix} \right)=\left( \begin{matrix} 0 & 2 & 2 \\ 3 & 3 & 5 \\ \end{matrix} \right)$$
As you can see, adding two matrices is the same as adding its elements. If we talk about multiplication, things change considerably. I recommend you this link, which is very instructive.
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