Matrix Operations: Step-By-Step Multiplication Example

7. Example:

   $AB = \begin{bmatrix}\boldsymbol{\color{green}{1}} & \boldsymbol{\color{green}{2}} \\ 3 & 4 \\ 5 & 6 \\ 7 & 8\end{bmatrix} \begin{bmatrix}1 & 0 & \boldsymbol{\color{green}{2}} \\ 0 & 1 & \boldsymbol{\color{green}{3}}\end{bmatrix} = \begin{bmatrix} 1 & 2 & \boldsymbol{\color{green}{8}} \\ c_{2,1} & c_{2,2} & c_{2,3}\\ c_{3,1} & c_{3,2} & c_{3,3}\\ c_{4,1} & c_{4,2} & c_{4,3} \end{bmatrix} = C$

   
   Explanation: We do $c_{1,3} =$$\; 1 \cdot 2 + 2 \cdot 3 =$$\; 2 + 6 =$$\; 8$. This is the element on row $1$ and column $3$, so we used row $1$ of matrix $A$ and column $3$ of matrix $B$.