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Question #51587

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $2n \times 2n$ skew-symmetric matrix with real entries. Assume that for any subset $I \subset \{1, 2, \dots, 2n\}$ with $|I| = n$ , the $n \times n$ submatrix $A_{I, I^c}$ (formed by the rows indexed by $I$ and columns indexed by $I^c = \{1, 2, \dots, 2n\} \setminus I$ ) is non-singular. Define $f(n) = \prod_{|I|=n} \det(A_{I, I^c})$ . Let $S_n \in \{+, -\}$ denote the sign of $f(n)$ . Which of the following represents the sequence $(S_1, S_2, S_3, S_4)$ ?

Options

A.
$(-, +, -, +)$
B.
$(-, +, +, +)$
C.
$(+, +, -, -)$
D.
$(-, -, +, +)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Set Complement Boundaries|Row/Column Exchange Parity|Determinant Product Rule

Why This Question Is Tricky

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Logical Assertion

Tests the causal relationship between two true statements.

Difficulty Boost : +4

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the properties of determinants under set complement boundaries and row/column parity shifts.

The core trap lies in the assumption that the sign of the product alternates with $n$ (i.e., $(-1)^n$ ).

Students correctly identify that swapping $I$ and $I^c$ introduces a $(-1)^n$ factor, but fail to account for the recursive logic that the total sign depends on the number of complementary pairs, $k = \frac{1}{2}\binom{2n}{n}$ .

Because $k$ grows non-linearly and is even for all $n \ge 2$ , the parity shift collapses, resulting in a positive product for $n=3$ despite $n$ being odd.

This exposes a lethal collision between local sign changes and global combinatorial parity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Set Complement Boundaries
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free