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I was just thinking that why we dont divide matrices by matrices, i understand that this is weird but i dont have a clear answer yet .

Is there any intuitive answer ? just like we cant have a dot product of a vector with a scalar .

Any kind of help is great.

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    In the field of real numbers $\mathbb R$, division of $a$ by $b$, ($b\neq 0$) is just $a.\frac{1}{b}$ so it is the operation multiplication what you misunderstood as a division. – 2017-01-07
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    Yeah, you are right. – 2017-01-07

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There is, but only in certain cases. As with real numbers, just define $$A\div B \equiv AB^{-1}$$ whenever this expression makes sense (i.e., when $B^{-1}$ exists and is compatible as a right multiplier of $A$).

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  1. Not all matrices are invertible.
  2. For invertible matrices, multiplication between them do not commute in general. So 'division' does not commute as well. Thus, we avoid the notion of division in a non-commutative ring.
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    There are 2 divisions - right division $A/B=AB^{-1},$ and left division $B\backslash A=B^{-1}A$ in the case if matrix $B$ is invertible – 2017-01-07
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    @bot, and MATLAB actually provides for both forms! – 2017-01-15