1. **Stating the problem:** We need to perform operations or solve a problem involving matrices. Since the exact question is not specified, let's assume a common matrix problem: multiplying two matrices. 2. **Formula and rules:** To multiply two matrices $A$ and $B$, the number of columns in $A$ must equal the number of rows in $B$. The element in the $i^{th}$ row and $j^{th}$ column of the product matrix $C$ is given by: $$C_{ij} = \sum_{k=1}^n A_{ik} \times B_{kj}$$ where $n$ is the number of columns in $A$ (or rows in $B$). 3. **Intermediate work:** Suppose $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \quad B = \begin{bmatrix} e & f \\ g & h \end{bmatrix}$$ Then, $$C = A \times B = \begin{bmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{bmatrix}$$ 4. **Explanation:** Each element of the product matrix is computed by multiplying elements of the rows of $A$ by corresponding elements of the columns of $B$ and summing the results. 5. **Final answer:** The product matrix is $$\begin{bmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{bmatrix}$$