1. Let's solve a matrix multiplication example. 2. Problem: Multiply matrices $$A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}$$ and $$B = \begin{bmatrix}5 & 6 \\ 7 & 8\end{bmatrix}$$. 3. Formula: The element in row $i$, column $j$ of the product matrix $C = AB$ is calculated as $$c_{ij} = \sum_{k} a_{ik} b_{kj}$$. 4. Calculate each element: - $$c_{11} = 1 \times 5 + 2 \times 7 = 5 + 14 = 19$$ - $$c_{12} = 1 \times 6 + 2 \times 8 = 6 + 16 = 22$$ - $$c_{21} = 3 \times 5 + 4 \times 7 = 15 + 28 = 43$$ - $$c_{22} = 3 \times 6 + 4 \times 8 = 18 + 32 = 50$$ 5. So, the product matrix is $$C = \begin{bmatrix}19 & 22 \\ 43 & 50\end{bmatrix}$$. This shows how to multiply two 2x2 matrices step-by-step.