Home / Linear algebra / Matrix multiplication
Example
Compute by hand:
\begin{flalign*} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} & = \begin{pmatrix} \vphantom{\Bigg(} \negthickspace \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 \\ 7 \end{pmatrix} & \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 6 \\ 8 \end{pmatrix} \negthickspace \end{pmatrix} \\ & = \begin{pmatrix} \negthickspace \vphantom{\Bigg(} 5 \begin{pmatrix} 1 \\ 3 \end{pmatrix} + 7 \begin{pmatrix} 2 \\ 4 \end{pmatrix} & 6 \begin{pmatrix} 1 \\ 3 \end{pmatrix} + 8 \begin{pmatrix} 2 \\ 4 \end{pmatrix} \negthickspace \end{pmatrix} \\ & = \begin{pmatrix} \negthickspace \vphantom{\Bigg(} \begin{pmatrix} 5 \\ 15 \end{pmatrix} + \begin{pmatrix} 14 \\ 28 \end{pmatrix} & \begin{pmatrix} 6 \\ 18 \end{pmatrix} + \begin{pmatrix} 16 \\ 32 \end{pmatrix} \negthickspace \end{pmatrix} \\ & = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}. & \end{flalign*}Compute with NumPy:
import numpy
a = numpy.array([[1, 2],
[3, 4]])
b = numpy.array([[5, 6],
[7, 8]])
a @ b
[[19 22] [43 50]]