1. **State the problem:** Find the matrices $2A + AB^T$ and $AB$ given $$A = \begin{pmatrix}5 & 2 & 3 \\ -1 & 0 & 4 \\ 7 & 2 & 1\end{pmatrix}, \quad B = \begin{pmatrix}3 & -2 & 1 \\ 4 & 7 & 2 \\ 1 & 5 & -3\end{pmatrix}.$$ 2. **Recall matrix operations:** - Scalar multiplication: multiply each element by the scalar. - Matrix multiplication: element $(i,j)$ of product is sum of products of row $i$ of first matrix and column $j$ of second. - Transpose $B^T$: swap rows and columns of $B$. 3. **Calculate $B^T$:** $$B^T = \begin{pmatrix}3 & 4 & 1 \\ -2 & 7 & 5 \\ 1 & 2 & -3\end{pmatrix}.$$ 4. **Calculate $AB^T$:** Multiply $A$ by $B^T$: $$AB^T = \begin{pmatrix}5 & 2 & 3 \\ -1 & 0 & 4 \\ 7 & 2 & 1\end{pmatrix} \times \begin{pmatrix}3 & 4 & 1 \\ -2 & 7 & 5 \\ 1 & 2 & -3\end{pmatrix}.$$ Calculate each element: - $(1,1): 5\times3 + 2\times(-2) + 3\times1 = 15 -4 +3 = 14$ - $(1,2): 5\times4 + 2\times7 + 3\times2 = 20 +14 +6 = 40$ - $(1,3): 5\times1 + 2\times5 + 3\times(-3) = 5 +10 -9 = 6$ - $(2,1): -1\times3 + 0\times(-2) + 4\times1 = -3 +0 +4 = 1$ - $(2,2): -1\times4 + 0\times7 + 4\times2 = -4 +0 +8 = 4$ - $(2,3): -1\times1 + 0\times5 + 4\times(-3) = -1 +0 -12 = -13$ - $(3,1): 7\times3 + 2\times(-2) + 1\times1 = 21 -4 +1 = 18$ - $(3,2): 7\times4 + 2\times7 + 1\times2 = 28 +14 +2 = 44$ - $(3,3): 7\times1 + 2\times5 + 1\times(-3) = 7 +10 -3 = 14$ So, $$AB^T = \begin{pmatrix}14 & 40 & 6 \\ 1 & 4 & -13 \\ 18 & 44 & 14\end{pmatrix}.$$ 5. **Calculate $2A$:** Multiply each element of $A$ by 2: $$2A = \begin{pmatrix}10 & 4 & 6 \\ -2 & 0 & 8 \\ 14 & 4 & 2\end{pmatrix}.$$ 6. **Calculate $2A + AB^T$:** Add corresponding elements: $$2A + AB^T = \begin{pmatrix}10+14 & 4+40 & 6+6 \\ -2+1 & 0+4 & 8+(-13) \\ 14+18 & 4+44 & 2+14\end{pmatrix} = \begin{pmatrix}24 & 44 & 12 \\ -1 & 4 & -5 \\ 32 & 48 & 16\end{pmatrix}.$$ 7. **Calculate $AB$:** Multiply $A$ by $B$: $$AB = \begin{pmatrix}5 & 2 & 3 \\ -1 & 0 & 4 \\ 7 & 2 & 1\end{pmatrix} \times \begin{pmatrix}3 & -2 & 1 \\ 4 & 7 & 2 \\ 1 & 5 & -3\end{pmatrix}.$$ Calculate each element: - $(1,1): 5\times3 + 2\times4 + 3\times1 = 15 +8 +3 = 26$ - $(1,2): 5\times(-2) + 2\times7 + 3\times5 = -10 +14 +15 = 19$ - $(1,3): 5\times1 + 2\times2 + 3\times(-3) = 5 +4 -9 = 0$ - $(2,1): -1\times3 + 0\times4 + 4\times1 = -3 +0 +4 = 1$ - $(2,2): -1\times(-2) + 0\times7 + 4\times5 = 2 +0 +20 = 22$ - $(2,3): -1\times1 + 0\times2 + 4\times(-3) = -1 +0 -12 = -13$ - $(3,1): 7\times3 + 2\times4 + 1\times1 = 21 +8 +1 = 30$ - $(3,2): 7\times(-2) + 2\times7 + 1\times5 = -14 +14 +5 = 5$ - $(3,3): 7\times1 + 2\times2 + 1\times(-3) = 7 +4 -3 = 8$ So, $$AB = \begin{pmatrix}26 & 19 & 0 \\ 1 & 22 & -13 \\ 30 & 5 & 8\end{pmatrix}.$$ **Final answers:** $$2A + AB^T = \begin{pmatrix}24 & 44 & 12 \\ -1 & 4 & -5 \\ 32 & 48 & 16\end{pmatrix}, \quad AB = \begin{pmatrix}26 & 19 & 0 \\ 1 & 22 & -13 \\ 30 & 5 & 8\end{pmatrix}.$$