1. **State the problem:** We are given two matrices $$A = \begin{pmatrix} -1 & 0 & -6 \\ 2 & 1 & 4 \\ 5 & 8 & -9 \end{pmatrix}$$ and $$B = \begin{pmatrix} 1 & -1 & 4 \\ 2 & 7 & -5 \\ 5 & 4 & 8 \end{pmatrix}$$ We need to find the matrix expression $$-2A + 7B$$. 2. **Recall the formula and rules:** Matrix scalar multiplication means multiplying every element of the matrix by the scalar. Matrix addition means adding corresponding elements. 3. **Calculate $$-2A$$:** $$-2A = -2 \times \begin{pmatrix} -1 & 0 & -6 \\ 2 & 1 & 4 \\ 5 & 8 & -9 \end{pmatrix} = \begin{pmatrix} -2 \times -1 & -2 \times 0 & -2 \times -6 \\ -2 \times 2 & -2 \times 1 & -2 \times 4 \\ -2 \times 5 & -2 \times 8 & -2 \times -9 \end{pmatrix} = \begin{pmatrix} 2 & 0 & 12 \\ -4 & -2 & -8 \\ -10 & -16 & 18 \end{pmatrix}$$ 4. **Calculate $$7B$$:** $$7B = 7 \times \begin{pmatrix} 1 & -1 & 4 \\ 2 & 7 & -5 \\ 5 & 4 & 8 \end{pmatrix} = \begin{pmatrix} 7 \times 1 & 7 \times -1 & 7 \times 4 \\ 7 \times 2 & 7 \times 7 & 7 \times -5 \\ 7 \times 5 & 7 \times 4 & 7 \times 8 \end{pmatrix} = \begin{pmatrix} 7 & -7 & 28 \\ 14 & 49 & -35 \\ 35 & 28 & 56 \end{pmatrix}$$ 5. **Add $$-2A$$ and $$7B$$:** $$-2A + 7B = \begin{pmatrix} 2 & 0 & 12 \\ -4 & -2 & -8 \\ -10 & -16 & 18 \end{pmatrix} + \begin{pmatrix} 7 & -7 & 28 \\ 14 & 49 & -35 \\ 35 & 28 & 56 \end{pmatrix} = \begin{pmatrix} 2+7 & 0+(-7) & 12+28 \\ -4+14 & -2+49 & -8+(-35) \\ -10+35 & -16+28 & 18+56 \end{pmatrix} = \begin{pmatrix} 9 & -7 & 40 \\ 10 & 47 & -43 \\ 25 & 12 & 74 \end{pmatrix}$$ **Final answer:** $$\boxed{\begin{pmatrix} 9 & -7 & 40 \\ 10 & 47 & -43 \\ 25 & 12 & 74 \end{pmatrix}}$$