Mathematical Methods In Economics

Step 1: Understanding quasiconcavity and quasiconvexity.
A function is quasiconcave if its upper level sets are convex. A function is quasiconvex if its lower level sets are convex. The relationships between these concepts can help identify the incorrect statement.

Step 2: Analysis of options.
- (A) If is quasiconcave, then is quasiconvex. This is correct because the negation of a quasiconcave function is quasiconvex.
- (B) If is a linear function, then it is quasiconcave as well as quasiconvex. This is correct because linear functions are both quasiconcave and quasiconvex.
- (C) Any concave function is quasiconcave, but the converse is not true. This is correct. A concave function is always quasiconcave, but not every quasiconcave function is concave.
- (D) Any convex function is quasiconcave, and its converse also holds. This is incorrect because not every convex function is quasiconcave. For instance, a convex function that is not concave may not be quasiconcave.

Step 3: Conclusion.
The incorrect statement is (D), which claims that the converse of convexity implies quasiconcavity, which is not true.

Step 1: Understanding symmetric matrices.
A matrix is symmetric if , i.e., it is equal to its transpose. In this question, both and are symmetric matrices.

Step 2: Analysis of options.
- (A) is a symmetric matrix: This is correct. The sum of two symmetric matrices is always symmetric.
- (B) is a symmetric matrix: This is correct. The sum of and is symmetric.
- (C) and are symmetric matrices: This is correct. Since and are symmetric, and are also symmetric.
- (D) is a symmetric matrix: This is incorrect. The difference is generally not symmetric, as matrix multiplication is not commutative.

Step 3: Conclusion.
The incorrect statement is (D), as is generally not a symmetric matrix.

Step 1: Matrix multiplication formula.
To find , multiply each element of the rows of by the corresponding elements of the columns of and sum them. Matrix multiplication is done as follows:

Step 2: Perform multiplication.
- For the first row and first column: - For the first row and second column: - For the second row and first column: - For the second row and second column: Thus, .

Step 3: Conclusion.
The correct matrix product is , so the answer is (A).

Step 1: Understanding the intervals.
- A **Closed Interval** includes both endpoints, so its representation is . This corresponds to option (I).
- An **Open Interval** excludes both endpoints, so its representation is . This corresponds to option (II).
- A **Unbounded Interval** is one where one endpoint is infinite, so its representation is . This corresponds to option (III).
- A **Half Open Interval** includes one endpoint and excludes the other, so its representation is . This corresponds to option (IV).

Step 2: Conclusion.
The correct match is (A) - (I), (B) - (II), (C) - (III), (D) - (IV).

- Quasiconcave: A function f(x) is quasiconcave if for any two points x₁ and x₂ in the domain, the line segment joining f(x₁) and f(x₂) lies entirely above the graph of the function.
- Quasiconvex: A function f(x) is quasiconvex if for any two points x₁ and x₂, the line segment joining f(x₁) and f(x₂) lies entirely below the graph of the function. • (A) If f(x) is quasiconcave, then -f(x) is quasiconvex: Correct. If a function is quasiconcave, then its negative is quasiconvex. • (B) If f(x) is a linear function, then it is quasiconcave as well as quasiconvex: Correct. A linear function is both quasiconcave and quasiconvex because it satisfies both the conditions for convex and concave functions. • (C) Any concave function is quasiconcave, but the converse is not true: Correct. Every concave function is quasiconcave, but not all quasiconcave functions are concave. • (D) Any convex function is quasiconvex and its converse also holds: Incorrect. While any convex function is quasiconvex, the converse is not true. There are quasiconvex functions that are not convex, making this statement false. The correct answer is (D) because the converse of the statement is not true: not every quasiconvex function is convex. Final Answer: Any convex function is quasiconvex and its converse also holds.