1. **State the problem:** We are asked to find the product of two matrices: $$A = \begin{bmatrix} 3 & 2 \\ 5 & -4 \\ -1 & -3 \end{bmatrix}$$ and $$B = \begin{bmatrix} 1 & 4 \\ -3 & 2 \end{bmatrix}$$ 2. **Recall the matrix multiplication rule:** The product $AB$ is defined if the number of columns in $A$ equals the number of rows in $B$. Here, $A$ is $3 \times 2$ and $B$ is $2 \times 2$, so multiplication is possible. 3. **Formula for matrix multiplication:** Each element $(i,j)$ of the product matrix $C = AB$ is computed as: $$c_{ij} = \sum_{k=1}^n a_{ik} b_{kj}$$ where $n$ is the number of columns in $A$ (or rows in $B$). 4. **Calculate each element:** - $c_{11} = 3 \times 1 + 2 \times (-3) = 3 - 6 = -3$ - $c_{12} = 3 \times 4 + 2 \times 2 = 12 + 4 = 16$ - $c_{21} = 5 \times 1 + (-4) \times (-3) = 5 + 12 = 17$ - $c_{22} = 5 \times 4 + (-4) \times 2 = 20 - 8 = 12$ - $c_{31} = (-1) \times 1 + (-3) \times (-3) = -1 + 9 = 8$ - $c_{32} = (-1) \times 4 + (-3) \times 2 = -4 - 6 = -10$ 5. **Write the product matrix:** $$AB = \begin{bmatrix} -3 & 16 \\ 17 & 12 \\ 8 & -10 \end{bmatrix}$$ This is the final answer.