1. **Stating the problem:** We want to multiply two matrices, say matrix $A$ of size $m \times n$ and matrix $B$ of size $n \times p$. The result will be a matrix $C$ of size $m \times p$. 2. **Formula and rules:** The element in the $i^{th}$ row and $j^{th}$ column of $C$, denoted $c_{ij}$, is calculated by the dot product of the $i^{th}$ row of $A$ and the $j^{th}$ column of $B$: $$ c_{ij} = \sum_{k=1}^n a_{ik} b_{kj} $$ Important rules: - The number of columns in $A$ must equal the number of rows in $B$. - Matrix multiplication is not commutative, so $AB \neq BA$ in general. 3. **Intermediate work example:** Suppose $$ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} $$ Calculate $C = AB$: - $c_{11} = 1 \times 5 + 2 \times 7 = 5 + 14 = 19$ - $c_{12} = 1 \times 6 + 2 \times 8 = 6 + 16 = 22$ - $c_{21} = 3 \times 5 + 4 \times 7 = 15 + 28 = 43$ - $c_{22} = 3 \times 6 + 4 \times 8 = 18 + 32 = 50$ So, $$ C = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix} $$ 4. **Explanation:** To multiply matrices, take each row of the first matrix and multiply it element-wise with each column of the second matrix, then sum those products to get each element of the result matrix. Repeat for all rows and columns. This process combines rows and columns to produce a new matrix representing the composition of linear transformations or systems.