Matrix - Matrix Multiplication

Matrix multiplication is only well-defined for matrices where the number of columns in the first matrix equals the number of rows in the second matrix.

To multiply two matrices $A$ and $B$, where $A$ is an $m$ by $n$ matrix and $B$ is an $n$ by $p$ matrix, the resulting matrix $C=AB$ will be an $m$ by $p$ matrix defined by

$$C_{i,j}=\sum _{k=1}^n{A}_{i,k}{B}_{k,j}\text{for }1\le i\le m,1\le j\le p$$

This means that each entry $C_{i,j}$ in the resulting matrix is computed by taking the Dot Product of the $i$-th row of matrix $A$ and the $j$-th column of matrix $B$.

Example

Consider the following matrices:

$$A=\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right),B=\left(\begin{array}{cc}7& 8\\ 9& 10\\ 11& 12\end{array}\right)$$

The product $C=AB$ is calculated as follows:

$$\begin{array}{rl}{C}_{1,1}& =1\cdot 7+2\cdot 9+3\cdot 11=58\\ C_{1,2}& =1\cdot 8+2\cdot 10+3\cdot 12=64\\ C_{2,1}& =4\cdot 7+5\cdot 9+6\cdot 11=139\\ C_{2,2}& =4\cdot 8+5\cdot 10+6\cdot 12=154\end{array}$$

Resulting in

$$C=\left(\begin{array}{cc}58& 64\\ 139& 154\end{array}\right)$$