Matrix Operations: Step-By-Step Multiplication Example

7. Example:

   $AB = \begin{bmatrix}1 & 2 \\ \boldsymbol{\color{green}{3}} & \boldsymbol{\color{green}{4}} \\ 5 & 6 \\ 7 & 8\end{bmatrix} \begin{bmatrix}1 & 0 & \boldsymbol{\color{green}{2}} \\ 0 & 1 & \boldsymbol{\color{green}{3}}\end{bmatrix} = \begin{bmatrix} 1 & 2 & 8 \\ 3 & 4 & \boldsymbol{\color{green}{18}}\\ c_{3,1} & c_{3,2} & c_{3,3}\\ c_{4,1} & c_{4,2} & c_{4,3} \end{bmatrix} = C$

   
   Explanation: We do $c_{2,3} =$$\; 3 \cdot 2 + 4 \cdot 3 =$$\; 6 + 12 =$$\; 18$. This is the element on row $2$ and column $3$, so we used row $2$ of matrix $A$ and column $3$ of matrix $B$.
   
   We'll stop here: try computing the other $6$ elements in a similar fashion.