Is there some sort of pattern to multiplying matrices?

Solution 1:

When multiplying, don't think of matrices as grids of numbers, think of the left matrix as a set of rows and the right matrix as a set of columns. Then you multiply rows on the left by columns on the right. How do you multiply a row and a column? You multiply corresponding elements then add

e.g $\left( \begin{matrix} 2 & 1 & -1 \\ \end{matrix} \right)\left( \begin{matrix} 3 \\ 2 \\ 1 \\ \end{matrix} \right)=2(3)+1(2)+(-1)(1)=7$

Note that this only makes sense if a row of the left matrix is the same size as a column of the right matrix. Where do these row times columns go? The product matrix will have as many rows as the left matrix (you are multiplying rows) and as many columns as the right matrix. The number in the second row and first column of the answer, for example, will be the product of the second row and the first column. Similarly for all the other entries in the product matrix.