1. **State the problem:** We need to find the product of matrices $B$ and $C$, where $$B = \begin{bmatrix} -3 & -5 \\ 0 & 8 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 & 1 \\ 6 & -2 & 3 \end{bmatrix}$$ 2. **Recall the matrix multiplication rule:** The product $BC$ is defined if the number of columns in $B$ equals the number of rows in $C$. Here, $B$ is $2 \times 2$ and $C$ is $2 \times 3$, so multiplication is possible. 3. **Formula for matrix multiplication:** The element in row $i$, column $j$ of the product matrix is $$ (BC)_{ij} = \sum_{k=1}^n B_{ik} C_{kj} $$ where $n$ is the number of columns in $B$ (or rows in $C$). 4. **Calculate each element:** - First row, first column: $$ (-3)(1) + (-5)(6) = -3 - 30 = -33 $$ - First row, second column: $$ (-3)(0) + (-5)(-2) = 0 + 10 = 10 $$ - First row, third column: $$ (-3)(1) + (-5)(3) = -3 - 15 = -18 $$ - Second row, first column: $$ (0)(1) + (8)(6) = 0 + 48 = 48 $$ - Second row, second column: $$ (0)(0) + (8)(-2) = 0 - 16 = -16 $$ - Second row, third column: $$ (0)(1) + (8)(3) = 0 + 24 = 24 $$ 5. **Write the resultant matrix:** $$ BC = \begin{bmatrix} -33 & 10 & -18 \\ 48 & -16 & 24 \end{bmatrix} $$ **Final answer:** $$ \boxed{\begin{bmatrix} -33 & 10 & -18 \\ 48 & -16 & 24 \end{bmatrix}} $$