1. **Problem Statement:** Given matrices $$A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}, \quad B = \begin{bmatrix}1 & 2 \\ -1 & 1\end{bmatrix}$$ Determine which statements are true: (i) $$(A^2)^T = (A^T)^2$$ (ii) $$(AB)^T = A^T B^T$$ 2. **Recall the properties of transpose:** - $$(XY)^T = Y^T X^T$$ for any matrices $X, Y$ where the product is defined. - $$(X^T)^2 = X^T X^T$$ means multiplying $X^T$ by itself. 3. **Calculate $A^2$:** $$A^2 = A \times A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix} \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix} = \begin{bmatrix}1\times1 + 2\times3 & 1\times2 + 2\times4 \\ 3\times1 + 4\times3 & 3\times2 + 4\times4\end{bmatrix} = \begin{bmatrix}7 & 10 \\ 15 & 22\end{bmatrix}$$ 4. **Calculate $(A^2)^T$:** Transpose swaps rows and columns: $$(A^2)^T = \begin{bmatrix}7 & 10 \\ 15 & 22\end{bmatrix}^T = \begin{bmatrix}7 & 15 \\ 10 & 22\end{bmatrix}$$ 5. **Calculate $A^T$:** $$A^T = \begin{bmatrix}1 & 3 \\ 2 & 4\end{bmatrix}$$ 6. **Calculate $(A^T)^2$:** $$A^T \times A^T = \begin{bmatrix}1 & 3 \\ 2 & 4\end{bmatrix} \begin{bmatrix}1 & 3 \\ 2 & 4\end{bmatrix} = \begin{bmatrix}1\times1 + 3\times2 & 1\times3 + 3\times4 \\ 2\times1 + 4\times2 & 2\times3 + 4\times4\end{bmatrix} = \begin{bmatrix}7 & 15 \\ 10 & 22\end{bmatrix}$$ 7. **Compare $(A^2)^T$ and $(A^T)^2$:** $$(A^2)^T = \begin{bmatrix}7 & 15 \\ 10 & 22\end{bmatrix}, \quad (A^T)^2 = \begin{bmatrix}7 & 15 \\ 10 & 22\end{bmatrix}$$ They are equal, so statement (i) is **true**. 8. **Calculate $AB$:** $$AB = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix} \begin{bmatrix}1 & 2 \\ -1 & 1\end{bmatrix} = \begin{bmatrix}1\times1 + 2\times(-1) & 1\times2 + 2\times1 \\ 3\times1 + 4\times(-1) & 3\times2 + 4\times1\end{bmatrix} = \begin{bmatrix}-1 & 4 \\ -1 & 10\end{bmatrix}$$ 9. **Calculate $(AB)^T$:** $$(AB)^T = \begin{bmatrix}-1 & 4 \\ -1 & 10\end{bmatrix}^T = \begin{bmatrix}-1 & -1 \\ 4 & 10\end{bmatrix}$$ 10. **Calculate $A^T B^T$:** $$B^T = \begin{bmatrix}1 & -1 \\ 2 & 1\end{bmatrix}$$ $$A^T B^T = \begin{bmatrix}1 & 3 \\ 2 & 4\end{bmatrix} \begin{bmatrix}1 & -1 \\ 2 & 1\end{bmatrix} = \begin{bmatrix}1\times1 + 3\times2 & 1\times(-1) + 3\times1 \\ 2\times1 + 4\times2 & 2\times(-1) + 4\times1\end{bmatrix} = \begin{bmatrix}7 & 2 \\ 10 & 2\end{bmatrix}$$ 11. **Compare $(AB)^T$ and $A^T B^T$:** $$(AB)^T = \begin{bmatrix}-1 & -1 \\ 4 & 10\end{bmatrix}, \quad A^T B^T = \begin{bmatrix}7 & 2 \\ 10 & 2\end{bmatrix}$$ They are not equal, so statement (ii) is **false**. 12. **Final conclusion:** (i) is true, (ii) is false. **Answer: B. (i) is true, (ii) is false.**