1. **State the problem:** We are given matrices $$A = \begin{bmatrix}-5 & -1 & 5 \\ 3 & -3 & -8\end{bmatrix}$$ and $$B = \begin{bmatrix}7 & -1 \\ -4 & -4 \\ -6 & 1\end{bmatrix}$$ We need to find the products $A \cdot B$ and $B \cdot A$. 2. **Recall matrix multiplication rules:** - The product $A \cdot B$ is defined if the number of columns of $A$ equals the number of rows of $B$. - The resulting matrix has dimensions equal to the number of rows of $A$ by the number of columns of $B$. Here, $A$ is $2 \times 3$ and $B$ is $3 \times 2$, so $A \cdot B$ is $2 \times 2$. Similarly, $B \cdot A$ is defined since $B$ is $3 \times 2$ and $A$ is $2 \times 3$, so $B \cdot A$ is $3 \times 3$. 3. **Calculate $A \cdot B$:** $$A \cdot B = \begin{bmatrix}-5 & -1 & 5 \\ 3 & -3 & -8\end{bmatrix} \cdot \begin{bmatrix}7 & -1 \\ -4 & -4 \\ -6 & 1\end{bmatrix}$$ Calculate each element: - Element (1,1): $(-5)(7) + (-1)(-4) + (5)(-6) = -35 + 4 - 30 = -61$ - Element (1,2): $(-5)(-1) + (-1)(-4) + (5)(1) = 5 + 4 + 5 = 14$ - Element (2,1): $(3)(7) + (-3)(-4) + (-8)(-6) = 21 + 12 + 48 = 81$ - Element (2,2): $(3)(-1) + (-3)(-4) + (-8)(1) = -3 + 12 - 8 = 1$ So, $$A \cdot B = \begin{bmatrix}-61 & 14 \\ 81 & 1\end{bmatrix}$$ 4. **Calculate $B \cdot A$:** $$B \cdot A = \begin{bmatrix}7 & -1 \\ -4 & -4 \\ -6 & 1\end{bmatrix} \cdot \begin{bmatrix}-5 & -1 & 5 \\ 3 & -3 & -8\end{bmatrix}$$ Calculate each element: - Element (1,1): $(7)(-5) + (-1)(3) = -35 - 3 = -38$ - Element (1,2): $(7)(-1) + (-1)(-3) = -7 + 3 = -4$ - Element (1,3): $(7)(5) + (-1)(-8) = 35 + 8 = 43$ - Element (2,1): $(-4)(-5) + (-4)(3) = 20 - 12 = 8$ - Element (2,2): $(-4)(-1) + (-4)(-3) = 4 + 12 = 16$ - Element (2,3): $(-4)(5) + (-4)(-8) = -20 + 32 = 12$ - Element (3,1): $(-6)(-5) + (1)(3) = 30 + 3 = 33$ - Element (3,2): $(-6)(-1) + (1)(-3) = 6 - 3 = 3$ - Element (3,3): $(-6)(5) + (1)(-8) = -30 - 8 = -38$ So, $$B \cdot A = \begin{bmatrix}-38 & -4 & 43 \\ 8 & 16 & 12 \\ 33 & 3 & -38\end{bmatrix}$$ **Final answers:** $$A \cdot B = \begin{bmatrix}-61 & 14 \\ 81 & 1\end{bmatrix}$$ $$B \cdot A = \begin{bmatrix}-38 & -4 & 43 \\ 8 & 16 & 12 \\ 33 & 3 & -38\end{bmatrix}$$