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I just came across this strange post where this fellow (is a one-by-one-matrix just a number (scalar)?) supposedly does bad maths and multiplies matrices of incompatible sizes by padding it with zeros as:

$$ \begin{bmatrix} c \end{bmatrix} \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} = \begin{bmatrix} c & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} = \begin{bmatrix} ca_{11} & ca_{12} & ca_{13} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} $$

Now my question is, why is this a bad idea? I might not be particularly useful, but it is a possible definition. Why is this not used?

user27221
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    People do sometimes pad matrices with zeros. The thing is, when you need the zeros you just write them into the matrix. So making yet another definition seems unnecessary. – David K Jun 06 '19 at 23:56
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    You can define anything you want. If it's not useful, it won't be used. – Gerry Myerson Jun 07 '19 at 01:04

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There exists the Kronecker product of matrices, which allows for the multiplication of two matrices of any size. There also exists the direct sum, which is an interesting way of adding matrices. These two other forms of matrix operations prove to be useful in certain circumstances. We are free to define operations as we please, it just happens to be the case that some definitions are more useful than others. If the operation that you have defined above is not used, it is because is not very useful; or at least, no one has found a useful purpose for it thus far.

Please note that there are likely more matrix operations that I have not covered. I have just listed what I know of.

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Kraigolas
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