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Question #41126

Difficulty: 8/10 | 🟢 Free

The Question

Let $M_n(\mathbb{R})$ be the set of all $n \times n$ real matrices ( $n \ge 2$ ). Define a mapping $f: M_n(\mathbb{R}) \to M_n(\mathbb{R})$ by $f(A) = A^T A$ . Let $V = \{A \in M_n(\mathbb{R}) \mid A^2 = I_n\}$ be the set of all involutory matrices, and let $W = f(V)$ be the image of $V$ under $f$ . Which of the following statements is ALWAYS true?

Options

A.
Every matrix in $W$ is similar to its own inverse.
B.
$W$ is the set of all symmetric positive definite matrices in $M_n(\mathbb{R})$ .
C.
The mapping $f: V \to W$ is injective.
D.
Every matrix in $W$ is an idempotent matrix.

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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

Surjective Mapping (Onto)|Idempotent and Involutory Properties|Transpose of Product

Why This Question Is Tricky

Common Mistakes

This question tests the properties of matrix mappings and the trap of assumed surjectivity.

Students often incorrectly assume that $f(A) = A^T A$ maps onto the entire space of symmetric positive definite (SPD) matrices when the domain is restricted to a specific class like involutory matrices.

By forcing the student to analyze the algebraic structure of $f(V)$ , the question reveals that the image $W$ is strictly restricted to SPD matrices whose eigenvalues occur in reciprocal pairs (since $B \sim B^{-1}$ ).

This shatters the false assumption of surjectivity and highlights the elegant interplay between involutory properties and similarity transformations.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Surjective Mapping (Onto)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free