1. **Problem Statement:** How to multiply two matrices. 2. **Matrix Multiplication Rule:** To multiply two matrices $A$ and $B$, the number of columns in $A$ must equal the number of rows in $B$. 3. **Formula:** If $A$ is an $m \times n$ matrix and $B$ is an $n \times p$ matrix, their product $C = AB$ is an $m \times p$ matrix where each element $c_{ij}$ is calculated as: $$ c_{ij} = \sum_{k=1}^n a_{ik} b_{kj} $$ This means each element in $C$ is the dot product of the $i$th row of $A$ and the $j$th column of $B$. 4. **Step-by-step example:** Suppose $$ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} $$ Calculate $C = AB$: - Element $c_{11} = 1 \times 5 + 2 \times 7 = 5 + 14 = 19$ - Element $c_{12} = 1 \times 6 + 2 \times 8 = 6 + 16 = 22$ - Element $c_{21} = 3 \times 5 + 4 \times 7 = 15 + 28 = 43$ - Element $c_{22} = 3 \times 6 + 4 \times 8 = 18 + 32 = 50$ So, $$ C = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix} $$ 5. **Summary:** Multiply rows of the first matrix by columns of the second matrix, summing the products to get each element of the result matrix.