https://file.aiquickdraw.com/mathbot/file/1726674012389ekeeis0l.jpg

Here are the answers to the questions based on the image:

27. **(d) None of the above**

28. **(d) \( A = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} \)**

29. **(d) Both A and B are null sets**

30. **(c) \( R = \{(1, 2), (2, 3)\} \)**

31. **(a) \( R = \{(3, 3), (4, 4), (6, 6)\} \)**

32. **(b) 128**

33. **(d) \(\{(1, 3), (1, 5), (3, 5)\}\)**

34. **(d) None of the above**

35. **(b) Circular relation**

36. **(c) \( K(T(x) + T(y)) = T(Kx + Ky) \) for all \( k \in \mathbb{R} \)**

37. **(c) \( f = \{(7, 8), (8, 9), (9, 9)\} \)**

38. **(c) \( x = 3, y = 9 \)**

39. **(e) None of the above**

40. **(a) Linearly dependent**

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### Relative Questions:
1. What is the process for solving a system of linear equations?
2. How do you determine if matrices are orthogonal?
3. What properties define a null set in set theory?
4. How can you identify whether a relation is reflexive, symmetric, or transitive?
5. What is the significance of linear dependence and independence in vector spaces?

**Tip:** Always verify the properties of sets and relations when working with abstract algebra and linear algebra problems to avoid confusion.

Solve the system of linear equations and answer the questions involving matrix operations, set theory, and functions as shown in the image.

Explore solutions to a system of linear equations and answer advanced questions involving matrix orthogonality, set theory, and linear transformations. Detailed step-by-step solutions suitable for undergraduate students in linear algebra and abstract algebra.

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