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I wanted to make the title: 'When is Matrix Multiplication Well-Defined', but the software wouldn't let me.

If we have two matrices $A$ and $B$, then $AB$ is well defined if and only if the number of columns of $A$ equals the number of rows of $B$.

Is this right?

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Yes, that’s exactly right. – Brian M. Scott Dec 8 '12 at 23:10
Great, thanks a lot! – matrix Dec 8 '12 at 23:12
My pleasure. $\,$ – Brian M. Scott Dec 8 '12 at 23:13

If $A=(a_{ij})$ and $B=(b_{jk})$ are matrices, with $1\leq i\leq n$, $1\leq j\leq p$, $1\leq k\leq q$ then the product $C=AB=(c_{ik})$ is the $(n\times q)$-matrix define by $c_{ik}=\sum_{j=1}^{p}a_{ij}b_{jk}$.

Here is an image from

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Nice picture! You should share it on Wikipedia ;) – Damian Sobota Dec 8 '12 at 23:29

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