Worked example

Compute

\[\begin{align*}
  Y
  = A B =
  \begin{pmatrix}
    1 & 2 \\
    3 & 4
  \end{pmatrix}
  \begin{pmatrix}
    5 & 6 \\
    7 & 8
  \end{pmatrix}
  = \begin{pmatrix} \vec{y}_1 & \vec{y}_2 \end{pmatrix}
  =
  \begin{pmatrix}
    19 & 22 \\
    43 & 50
  \end{pmatrix}
\end{align*}
\]

with

\[\begin{align*}
\vec{y}_1
& =
    \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}
    \begin{pmatrix} 5 \\ 7 \end{pmatrix}
=
    5 \begin{pmatrix} 1 \\ 3 \end{pmatrix}
    + 7 \begin{pmatrix} 2 \\ 4 \end{pmatrix}
=
      \begin{pmatrix} 5 \\ 15 \end{pmatrix}
    + \begin{pmatrix} 14 \\ 28 \end{pmatrix}
=
    \begin{pmatrix}
      19 \\ 43
    \end{pmatrix}
\\[1ex]
\vec{y}_2
& =
    \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}
    \begin{pmatrix} 6 \\ 8 \end{pmatrix}
=
    6 \begin{pmatrix} 1 \\ 3 \end{pmatrix}
    + 8 \begin{pmatrix} 2 \\ 4 \end{pmatrix}
=
    \begin{pmatrix} 6 \\ 18 \end{pmatrix}
    + \begin{pmatrix} 16 \\ 32 \end{pmatrix}
=
\begin{pmatrix}
22 \\ 50
\end{pmatrix}
\end{align*}
\]

by

import numpy as np

A = np.array([[1, 2],
              [3, 4]])

B = np.array([[5, 6],
              [7, 8]])

A @ B