1. Let's start with a basic problem: Find the product of two matrices. 2. Problem: Given matrices $$A = \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$$ and $$B = \begin{bmatrix}5 & 6\\7 & 8\end{bmatrix}$$, find the matrix product $$AB$$. 3. Formula: The product of two matrices $$A$$ and $$B$$, where $$A$$ is of size $$m \times n$$ and $$B$$ is of size $$n \times p$$, is a matrix $$C$$ of size $$m \times p$$ where each element $$c_{ij}$$ is calculated as: $$ c_{ij} = \sum_{k=1}^n a_{ik} b_{kj} $$ 4. Step-by-step calculation: - Calculate element $$c_{11}$$: $$1 \times 5 + 2 \times 7 = 5 + 14 = 19$$ - Calculate element $$c_{12}$$: $$1 \times 6 + 2 \times 8 = 6 + 16 = 22$$ - Calculate element $$c_{21}$$: $$3 \times 5 + 4 \times 7 = 15 + 28 = 43$$ - Calculate element $$c_{22}$$: $$3 \times 6 + 4 \times 8 = 18 + 32 = 50$$ 5. So, the product matrix $$AB$$ is: $$ \begin{bmatrix}19 & 22\\43 & 50\end{bmatrix} $$ This shows how to multiply two matrices by taking the dot product of rows of the first matrix with columns of the second matrix.