Matrix Operations: Step-By-Step Multiplication Example

7. Example:

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

   
   Explanation: We do $c_{1,2} =$$\; 1 \cdot 0 + 2 \cdot 1 =$$\; 0 + 2 =$$\; 2$. This is the element on row $1$ and column $2$, so we used row $1$ of matrix $A$ and column $2$ of matrix $B$.