1. **Problem:** Find the product of matrices $A$ and $B$ where $$A=\begin{pmatrix}1 & 2 & 0 & 3 \\ 2 & 2 & 6 & -9 \\ 1 & -1 & 2 & 1\end{pmatrix},\quad B=\begin{pmatrix}2 & 2 & 5 \\ 0 & 2 & 1 \\ 2 & 0 & -5 \\ 0 & -6 & 3\end{pmatrix}$$ 2. **Formula:** The product $AB$ is computed by multiplying rows of $A$ by columns of $B$: $$ (AB)_{ij} = \sum_{k=1}^4 A_{ik} B_{kj} $$ 3. **Calculation:** - First row: - $(1)(2)+(2)(0)+(0)(2)+(3)(0)=2+0+0+0=2$ - $(1)(2)+(2)(2)+(0)(0)+(3)(-6)=2+4+0-18=-12$ - $(1)(5)+(2)(1)+(0)(-5)+(3)(3)=5+2+0+9=16$ - Second row: - $(2)(2)+(2)(0)+(6)(2)+(-9)(0)=4+0+12+0=16$ - $(2)(2)+(2)(2)+(6)(0)+(-9)(-6)=4+4+0+54=62$ - $(2)(5)+(2)(1)+(6)(-5)+(-9)(3)=10+2-30-27=-45$ - Third row: - $(1)(2)+(-1)(0)+(2)(2)+(1)(0)=2+0+4+0=6$ - $(1)(2)+(-1)(2)+(2)(0)+(1)(-6)=2-2+0-6=-6$ - $(1)(5)+(-1)(1)+(2)(-5)+(1)(3)=5-1-10+3=-3$ 4. **Answer:** $$AB=\begin{pmatrix}2 & -12 & 16 \\ 16 & 62 & -45 \\ 6 & -6 & -3\end{pmatrix}$$