**Matrix multiplication** is an [[operation]] that takes an ordered pair $(A,B)$ of *compatible* [[matrix|matrices]] and outputs another matrix $M$. Let $A\in K^{m\times n}$ Let $B\in K^{n\times s}$ Let $M\in K^{m\times s}$ be the product of $A$ and $B$ (in this order) $M=AB=(m_{ik})$ $\begin{aligned} m_{ik}&=\mathbf a_{i©}\cdot\mathbf b_{®k}\\ &=\sum_{j=1}^{n} a_{i j} b_{j k}\\ &=a_{i 1} b_{1 k}+a_{i 2} b_{2 k}+\cdots+a_{i n} b_{n k} \end{aligned}$ $M=A B=\left(\begin{array}{ccc}\mathbf a_{1©}\cdot\mathbf b_{®1} & \cdots & \mathbf a_{1©}\cdot\mathbf b_{®s} \\ \vdots & \ddots& \vdots \\ \mathbf a_{m©}\cdot\mathbf b_{®1} & \cdots & \mathbf a_{m©}\cdot\mathbf b_{®s}\end{array}\right)$ - *Compatibility* is equivalent to equality of: - the number of columns, $n,$ from the first (left) matrix $A$ - the number of rows, $n,$ from the second (right) matrix $B$ - Matrices need to be compatible, because the product's elements $m_{ik}$ are defined using: - a [[dot product]] $\mathbf a_{i©}\cdot\mathbf b_{®k}$ of the corresponding - $i^{\text{th}}$ row of $A$ - written here as $\mathbf a_{i©}$ - $j^{\text{th}}$ row of $B$ - written here as $\mathbf b_{®k}$ - and these two vectors need to be from the same [[vector space]]