1. **State the problem:** Multiply the given matrices. **Problem 6:** Multiply the 3x3 matrices $$A = \begin{bmatrix}-6 & 1 & 1 \\ -2 & 3 & 8 \\ 1 & 7 & 1\end{bmatrix}$$ and $$B = \begin{bmatrix}0 & -1 & 3 \\ -7 & -2 & 4 \\ -1 & 3 & 4\end{bmatrix}$$ 2. **Recall matrix multiplication rule:** The element in row $i$, column $j$ of the product matrix $C = AB$ is $$c_{ij} = \sum_{k=1}^3 a_{ik} b_{kj}$$ 3. **Calculate each element of $C$: ** - $c_{11} = (-6)(0) + (1)(-7) + (1)(-1) = 0 -7 -1 = -8$ - $c_{12} = (-6)(-1) + (1)(-2) + (1)(3) = 6 -2 +3 = 7$ - $c_{13} = (-6)(3) + (1)(4) + (1)(4) = -18 +4 +4 = -10$ - $c_{21} = (-2)(0) + (3)(-7) + (8)(-1) = 0 -21 -8 = -29$ - $c_{22} = (-2)(-1) + (3)(-2) + (8)(3) = 2 -6 +24 = 20$ - $c_{23} = (-2)(3) + (3)(4) + (8)(4) = -6 +12 +32 = 38$ - $c_{31} = (1)(0) + (7)(-7) + (1)(-1) = 0 -49 -1 = -50$ - $c_{32} = (1)(-1) + (7)(-2) + (1)(3) = -1 -14 +3 = -12$ - $c_{33} = (1)(3) + (7)(4) + (1)(4) = 3 +28 +4 = 35$ 4. **Write the product matrix:** $$C = \begin{bmatrix}-8 & 7 & -10 \\ -29 & 20 & 38 \\ -50 & -12 & 35\end{bmatrix}$$ --- **Problem 8:** Multiply the 3x2 matrix $$D = \begin{bmatrix}1 & -4 \\ 3 & 0 \\ 2 & 2\end{bmatrix}$$ by the 2x3 matrix $$E = \begin{bmatrix}1 & 0 & 4 \\ 3 & 2 & 2\end{bmatrix}$$ 5. **Matrix multiplication rule:** The product $F = DE$ is a 3x3 matrix with elements $$f_{ij} = \sum_{k=1}^2 d_{ik} e_{kj}$$ 6. **Calculate each element of $F$: ** - $f_{11} = (1)(1) + (-4)(3) = 1 -12 = -11$ - $f_{12} = (1)(0) + (-4)(2) = 0 -8 = -8$ - $f_{13} = (1)(4) + (-4)(2) = 4 -8 = -4$ - $f_{21} = (3)(1) + (0)(3) = 3 + 0 = 3$ - $f_{22} = (3)(0) + (0)(2) = 0 + 0 = 0$ - $f_{23} = (3)(4) + (0)(2) = 12 + 0 = 12$ - $f_{31} = (2)(1) + (2)(3) = 2 + 6 = 8$ - $f_{32} = (2)(0) + (2)(2) = 0 + 4 = 4$ - $f_{33} = (2)(4) + (2)(2) = 8 + 4 = 12$ 7. **Write the product matrix:** $$F = \begin{bmatrix}-11 & -8 & -4 \\ 3 & 0 & 12 \\ 8 & 4 & 12\end{bmatrix}$$