1. **Problem 1: Matrix operations with 2x2 matrices** Given matrices: $$A=\begin{pmatrix}1 & 3 \\ -2 & 3\end{pmatrix}, B=\begin{pmatrix}2 & -1 \\ 1 & 4\end{pmatrix}, C=\begin{pmatrix}1 & 2 \\ 2 & 4\end{pmatrix}$$ We need to compute: a. $A+B$ b. $A+C$ c. $A-B$ d. $A-C$ **Formula for matrix addition/subtraction:** $$ (X \pm Y)_{ij} = X_{ij} \pm Y_{ij} $$ where $X_{ij}$ and $Y_{ij}$ are elements of matrices $X$ and $Y$ at row $i$, column $j$. **Step-by-step calculations:** 1. a. $A+B = \begin{pmatrix}1+2 & 3+(-1) \\ -2+1 & 3+4\end{pmatrix} = \begin{pmatrix}3 & 2 \\ -1 & 7\end{pmatrix}$ 2. b. $A+C = \begin{pmatrix}1+1 & 3+2 \\ -2+2 & 3+4\end{pmatrix} = \begin{pmatrix}2 & 5 \\ 0 & 7\end{pmatrix}$ 3. c. $A-B = \begin{pmatrix}1-2 & 3-(-1) \\ -2-1 & 3-4\end{pmatrix} = \begin{pmatrix}-1 & 4 \\ -3 & -1\end{pmatrix}$ 4. d. $A-C = \begin{pmatrix}1-1 & 3-2 \\ -2-2 & 3-4\end{pmatrix} = \begin{pmatrix}0 & 1 \\ -4 & -1\end{pmatrix}$ 2. **Problem 2: Matrix operations with 3x3 matrices** Given matrices: $$A=\begin{pmatrix}1 & 2 & 3 \\ -2 & 0 & 1 \\ 4 & 5 & 6\end{pmatrix}, B=\begin{pmatrix}3 & 1 & 4 \\ 4 & 0 & -1 \\ 5 & 2 & -2\end{pmatrix}$$ We need to compute: a. $A+B$ b. $A-B$ c. $B-A$ d. $A+0$ e. $0+B$ **Note:** The zero matrix $0$ here is the $3 \times 3$ zero matrix: $$0=\begin{pmatrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}$$ **Step-by-step calculations:** 1. a. $A+B = \begin{pmatrix}1+3 & 2+1 & 3+4 \\ -2+4 & 0+0 & 1+(-1) \\ 4+5 & 5+2 & 6+(-2)\end{pmatrix} = \begin{pmatrix}4 & 3 & 7 \\ 2 & 0 & 0 \\ 9 & 7 & 4\end{pmatrix}$ 2. b. $A-B = \begin{pmatrix}1-3 & 2-1 & 3-4 \\ -2-4 & 0-0 & 1-(-1) \\ 4-5 & 5-2 & 6-(-2)\end{pmatrix} = \begin{pmatrix}-2 & 1 & -1 \\ -6 & 0 & 2 \\ -1 & 3 & 8\end{pmatrix}$ 3. c. $B-A = \begin{pmatrix}3-1 & 1-2 & 4-3 \\ 4-(-2) & 0-0 & -1-1 \\ 5-4 & 2-5 & -2-6\end{pmatrix} = \begin{pmatrix}2 & -1 & 1 \\ 6 & 0 & -2 \\ 1 & -3 & -8\end{pmatrix}$ 4. d. $A+0 = A = \begin{pmatrix}1 & 2 & 3 \\ -2 & 0 & 1 \\ 4 & 5 & 6\end{pmatrix}$ 5. e. $0+B = B = \begin{pmatrix}3 & 1 & 4 \\ 4 & 0 & -1 \\ 5 & 2 & -2\end{pmatrix}$ **Final answers:** Problem 1: - a. $\begin{pmatrix}3 & 2 \\ -1 & 7\end{pmatrix}$ - b. $\begin{pmatrix}2 & 5 \\ 0 & 7\end{pmatrix}$ - c. $\begin{pmatrix}-1 & 4 \\ -3 & -1\end{pmatrix}$ - d. $\begin{pmatrix}0 & 1 \\ -4 & -1\end{pmatrix}$ Problem 2: - a. $\begin{pmatrix}4 & 3 & 7 \\ 2 & 0 & 0 \\ 9 & 7 & 4\end{pmatrix}$ - b. $\begin{pmatrix}-2 & 1 & -1 \\ -6 & 0 & 2 \\ -1 & 3 & 8\end{pmatrix}$ - c. $\begin{pmatrix}2 & -1 & 1 \\ 6 & 0 & -2 \\ 1 & -3 & -8\end{pmatrix}$ - d. $\begin{pmatrix}1 & 2 & 3 \\ -2 & 0 & 1 \\ 4 & 5 & 6\end{pmatrix}$ - e. $\begin{pmatrix}3 & 1 & 4 \\ 4 & 0 & -1 \\ 5 & 2 & -2\end{pmatrix}$