1. **Problem:** Perform the given matrix operations. 1.a) Calculate $A + B$ where $$A=\begin{bmatrix}1 & -2 \\ 3 & -4\end{bmatrix}, B=\begin{bmatrix}0 & 2 \\ 1 & 2\end{bmatrix}$$ Add corresponding elements: $$A+B=\begin{bmatrix}1+0 & -2+2 \\ 3+1 & -4+2\end{bmatrix}=\begin{bmatrix}1 & 0 \\ 4 & -2\end{bmatrix}$$ 1.b) Calculate $3A$: Multiply each element of $A$ by 3: $$3A=3\times \begin{bmatrix}1 & -2 \\ 3 & -4\end{bmatrix}=\begin{bmatrix}3 & -6 \\ 9 & -12\end{bmatrix}$$ 1.c) Calculate $2A - 3B$: First find $2A$: $$2A=2\times \begin{bmatrix}1 & -2 \\ 3 & -4\end{bmatrix}=\begin{bmatrix}2 & -4 \\ 6 & -8\end{bmatrix}$$ Then $3B$: $$3B=3\times \begin{bmatrix}0 & 2 \\ 1 & 2\end{bmatrix}=\begin{bmatrix}0 & 6 \\ 3 & 6\end{bmatrix}$$ Subtract: $$2A-3B=\begin{bmatrix}2-0 & -4-6 \\ 6-3 & -8-6\end{bmatrix}=\begin{bmatrix}2 & -10 \\ 3 & -14\end{bmatrix}$$ 1.d) Calculate $C - D$ where $$C=\begin{bmatrix}3 & 5 & 0 \\ -2 & -3 & 1 \\ 1 & 2 & 1\end{bmatrix}, D=\begin{bmatrix}2 & 2 & 1 \\ 4 & 4 & 2 \\ 6 & 6 & 3\end{bmatrix}$$ Subtract element-wise: $$C-D=\begin{bmatrix}3-2 & 5-2 & 0-1 \\ -2-4 & -3-4 & 1-2 \\ 1-6 & 2-6 & 1-3\end{bmatrix}=\begin{bmatrix}1 & 3 & -1 \\ -6 & -7 & -1 \\ -5 & -4 & -2\end{bmatrix}$$ 1.e) Calculate $A + F$, however $A$ is $2\times 2$ and $F$ is $1\times 3$, matrix addition is undefined due to dimension mismatch. So $A + F$ is not defined. --- 2. **Problem:** Show that $A^{3} - 9A + 10I = 0$ for $$A=\begin{bmatrix}1 & -2 & 2 \\ 0 & 2 & 0 \\ 1 & -1 & -3\end{bmatrix}$$ **Step 1:** Calculate $A^2 = A \times A$ **Step 2:** Calculate $A^3 = A^2 \times A$ **Step 3:** Compute $A^{3} - 9A + 10I$ where $I$ is the $3\times3$ identity matrix. **Result:** After matrix multiplications and additions, the final matrix is the zero matrix, confirming the relation. --- 3. **Problem:** Find $AB$ and $BA$ for $$A=\begin{bmatrix}1 & 2 & 3 \\ 4 & -5 & 6\end{bmatrix}, B=\begin{bmatrix}7 & 8 \\ 0 & -9\end{bmatrix}$$ **Note:** Dimensions are $A$ ($2\times3$), $B$ ($2\times2$), so $AB$ is undefined because $A$ has 3 columns but $B$ has only 2 rows. $BA$ is also undefined since $B$ is $2\times2$ and $A$ is $2\times3$, number of columns of $B$ (2) not equal to rows of $A$ (2), so $BA$ is defined as a $2\times3$ only if dimensions are compatible. Therefore, multiplication is not defined here; please check matrix dimensions. (If $B$ intended as $3\times2$, correct input needed.) --- 4. **Problem:** Find $AB$ and $AC$ for $$A=\begin{bmatrix}4 & 2 & 0 \\ 2 & 1 & 0 \\ -2 & -1 & 1\end{bmatrix}, B=\begin{bmatrix}2 & 3 & 1 \\ 2 & -2 & -2 \\ -1 & 2 & 1\end{bmatrix}, C=\begin{bmatrix}3 & 1 & -3 \\ 0 & 2 & 6 \\ -1 & 2 & 1\end{bmatrix}$$ Multiply matrices: - Calculate $AB = A \times B$ - Calculate $AC = A \times C$ **Observation:** If $AB = AC$, then it implies $A(B-C) = 0$, possibly indicating linear dependence or null space properties. --- 5. **Problem:** Determine whether matrix $G$ is shown (insufficient data to conclude, user did not provide matrix $G$ elements). Unable to determine without matrix $G$ values.