Matrix/Multiplication/Definition/Explanations/Remark

A matrix multiplication is only possible when the number of columns of the left-hand matrix equals the number of rows of the right-hand matrix. Just think of the scheme

${}(ROWROW){\begin{pmatrix}C\\O\\L\\U\\M\\N\end{pmatrix}}=(RC+O^{2}+WL+RU+OM+WN)\,,$

the result is an ${}1\times 1$-Matrix. In particular, one can multiply an ${}m\times n$-matrix ${}A$ with a column vector of length ${}n$ (the vector on the right), and the result is a column vector of length ${}m$. The two matrices can also be multiplied with roles interchanged,

${}{\begin{pmatrix}C\\O\\L\\U\\M\\N\end{pmatrix}}(ROWROW)={\begin{pmatrix}CR&CO&CW&CR&CO&CW\\OR&O^{2}&OW&OR&O^{2}&OW\\LR&LO&LW&LR&LO&LW\\UR&UO&UW&UR&UO&UW\\MR&MO&MW&MR&MO&MW\\NR&NO&NW&NR&NO&NW\end{pmatrix}}\,.$