1. **Problem Statement:** Given matrices $$A = \begin{bmatrix}-1 & 2 & 0 \\ 3 & -5 & 1 \\ -1 & 2 & 0\end{bmatrix}$$ and $$B = \begin{bmatrix}2 & 1 & -4 \\ -1 & 5 & 3 \\ -1 & 2 & -5\end{bmatrix}$$ Calculate: a. $2A + B$ b. $-3B + A$ c. $A - 2B^T$ --- 2. **Formulas and Rules:** - Matrix addition/subtraction: add/subtract corresponding elements. - Scalar multiplication: multiply each element by the scalar. - Transpose $B^T$: swap rows and columns of $B$. --- 3. **Calculations:** **a. Calculate $2A + B$:** First, compute $2A$: $$2A = 2 \times \begin{bmatrix}-1 & 2 & 0 \\ 3 & -5 & 1 \\ -1 & 2 & 0\end{bmatrix} = \begin{bmatrix}-2 & 4 & 0 \\ 6 & -10 & 2 \\ -2 & 4 & 0\end{bmatrix}$$ Now add $B$: $$2A + B = \begin{bmatrix}-2 & 4 & 0 \\ 6 & -10 & 2 \\ -2 & 4 & 0\end{bmatrix} + \begin{bmatrix}2 & 1 & -4 \\ -1 & 5 & 3 \\ -1 & 2 & -5\end{bmatrix} = \begin{bmatrix}-2+2 & 4+1 & 0+(-4) \\ 6+(-1) & -10+5 & 2+3 \\ -2+(-1) & 4+2 & 0+(-5)\end{bmatrix} = \begin{bmatrix}0 & 5 & -4 \\ 5 & -5 & 5 \\ -3 & 6 & -5\end{bmatrix}$$ **b. Calculate $-3B + A$:** First, compute $-3B$: $$-3B = -3 \times \begin{bmatrix}2 & 1 & -4 \\ -1 & 5 & 3 \\ -1 & 2 & -5\end{bmatrix} = \begin{bmatrix}-6 & -3 & 12 \\ 3 & -15 & -9 \\ 3 & -6 & 15\end{bmatrix}$$ Now add $A$: $$-3B + A = \begin{bmatrix}-6 & -3 & 12 \\ 3 & -15 & -9 \\ 3 & -6 & 15\end{bmatrix} + \begin{bmatrix}-1 & 2 & 0 \\ 3 & -5 & 1 \\ -1 & 2 & 0\end{bmatrix} = \begin{bmatrix}-6+(-1) & -3+2 & 12+0 \\ 3+3 & -15+(-5) & -9+1 \\ 3+(-1) & -6+2 & 15+0\end{bmatrix} = \begin{bmatrix}-7 & -1 & 12 \\ 6 & -20 & -8 \\ 2 & -4 & 15\end{bmatrix}$$ **c. Calculate $A - 2B^T$:** First, find transpose of $B$: $$B^T = \begin{bmatrix}2 & -1 & -1 \\ 1 & 5 & 2 \\ -4 & 3 & -5\end{bmatrix}$$ Then compute $2B^T$: $$2B^T = 2 \times \begin{bmatrix}2 & -1 & -1 \\ 1 & 5 & 2 \\ -4 & 3 & -5\end{bmatrix} = \begin{bmatrix}4 & -2 & -2 \\ 2 & 10 & 4 \\ -8 & 6 & -10\end{bmatrix}$$ Now subtract: $$A - 2B^T = \begin{bmatrix}-1 & 2 & 0 \\ 3 & -5 & 1 \\ -1 & 2 & 0\end{bmatrix} - \begin{bmatrix}4 & -2 & -2 \\ 2 & 10 & 4 \\ -8 & 6 & -10\end{bmatrix} = \begin{bmatrix}-1-4 & 2-(-2) & 0-(-2) \\ 3-2 & -5-10 & 1-4 \\ -1-(-8) & 2-6 & 0-(-10)\end{bmatrix} = \begin{bmatrix}-5 & 4 & 2 \\ 1 & -15 & -3 \\ 7 & -4 & 10\end{bmatrix}$$ --- **Final answers:** a. $2A + B = \begin{bmatrix}0 & 5 & -4 \\ 5 & -5 & 5 \\ -3 & 6 & -5\end{bmatrix}$ b. $-3B + A = \begin{bmatrix}-7 & -1 & 12 \\ 6 & -20 & -8 \\ 2 & -4 & 15\end{bmatrix}$ c. $A - 2B^T = \begin{bmatrix}-5 & 4 & 2 \\ 1 & -15 & -3 \\ 7 & -4 & 10\end{bmatrix}$