The general idea is to use a special operation called the dot product. Given two lists of numbers, where the lists are of equal length, you multiply corresponding elements in each list, then add the products together.
For example, consider the lists {1, 2, 3} and {0, -1, 2}. The dot product of these two lists is 1*0 + 2*(-1) + 3*2 = 0 - 2 + 6 = 4. In matrix form, we would denote this example by
[tex]\begin{bmatrix}1&2&3\end{bmatrix}\begin{bmatrix}0\\-1\\2\end{bmatrix}=4[/tex]
In either case, the orientation of one matrix has to be the opposite of the other, so that the number of rows in the first matrix matches the number of columns in the second matrix.
For matrices of larger size, we can break up the whole product into smaller, 1-row/1-column products like in the example above. For instance, consider the matrix product
[tex]\begin{bmatrix}1&0&-1\\-1&2&0\end{bmatrix}\begin{bmatrix}2&3\\-2&0\\1&1\end{bmatrix}[/tex]
The elements in the resulting matrix correspond to the dot products of different combinations of rows from the first matrix and columns from the second matrix. We have
[tex]\begin{bmatrix}1&0&-1\\-1&2&0\end{bmatrix}\begin{bmatrix}2&3\\-2&0\\1&1\end{bmatrix}=\begin{bmatrix}\begin{bmatrix}1&0&-1\end{bmatrix}\begin{bmatrix}2\\-2\\1\end{bmatrix}&\begin{bmatrix}1&0&-1\end{bmatrix}\begin{bmatrix}3\\0\\1\end{bmatrix}\\\\\begin{bmatrix}-1&2&0\end{bmatrix}\begin{bmatrix}2\\-2\\1\end{bmatrix}&\begin{bmatrix}-1&2&0\end{bmatrix}\begin{bmatrix}3\\0\\1\end{bmatrix}\end{bmatrix}=\begin{bmatrix}1&2\\-6&-3\end{bmatrix}[/tex]
