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Question #41099

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{ A \in M_3(\mathbb{R}) \mid A^T A = I \text{ and } \det(A) = 1 \}$ . For any matrix $M \in M_3(\mathbb{R})$ , its Frobenius norm is defined as $||M|| = \sqrt{\text{Tr}(M^T M)}$ . Two relations $\mathcal{R}_1$ and $\mathcal{R}_2$ are defined on $S$ as follows: $A \ \mathcal{R}_1 \ B \iff ||A - B||^2 < p$ $A \ \mathcal{R}_2 \ B \iff ||A - B||^2 > q$ If $p$ is the smallest integer such that $\mathcal{R}_1$ is the universal relation on $S$ , and $q$ is the smallest integer such that $\mathcal{R}_2$ is the empty relation on $S$ , then the value of $p + q$ is:

Options

A.
17
B.
25
C.
16
D.
24

Check Your Answer

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

Empty and Universal Relation|Orthogonal Matrix Conditions|Modulus Inequality Logic|Geometrical Interpretation of Modulus

Why This Question Is Tricky

Common Mistakes

The core trap lies in the geometric interpretation of the maximum distance between two orthogonal matrices.

Students often assume the maximum distance occurs when $B = -A$ , which yields $||A - B||^2 = 12$ for $3 \times 3$ matrices, leading to the incorrect sum of 25.

However, the domain is restricted to $SO(3)$ (matrices with determinant 1).

If $A \in SO(3)$ , then $-A$ has determinant $-1$ , meaning $-A \notin SO(3)$ .

By analyzing the eigenvalues of the transition matrix $A^T B \in SO(3)$ , the true maximum squared distance is revealed to be 8.

The problem elegantly combines this linear algebra trap with the set-theoretic definitions of universal and empty relations via metric bounds.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Empty and Universal Relation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free