Matrix Multiplication

37. Matrix Multiplication

For matrices $𝐴$ ( $𝑚 \times 𝑛$ ) and $𝐵$ ( $𝑛 \times 𝑝$ ), the product $𝐶 = 𝐴 𝐵$ is the $𝑚 \times 𝑝$ matrix with entries:

𝑐_{𝑖 𝑗} = \sum_{𝑘 = 1}^{𝑛} 𝑎_{𝑖 𝑘} 𝑏_{𝑘 𝑗}

Each entry $𝑐_{𝑖 𝑗}$ is the dot product of row $𝑖$ of $𝐴$ with column $𝑗$ of $𝐵$ .

Compatibility: inner dimensions must match — $𝐴$ ’s columns must equal $𝐵$ ’s rows.

Example

𝐴 = [\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] 𝐵 = [\begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix}]

𝐴 𝐵 = [\begin{matrix} 1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \\ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{matrix}] = [\begin{matrix} 19 & 22 \\ 43 & 50 \end{matrix}]

If $𝐴$ and $𝐵$ represent linear transformations, then $𝐴 𝐵$ represents the composition — apply $𝐵$ first, then $𝐴$ (see Composition of Linear Transformations).