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Question #51367

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ be a $3 \times 3$ non-scalar matrix with real entries such that $A^2 = A$ and $\text{trace}(A) = 2$ . Let $B = I_3 - 2A$ , where $I_3$ is the $3 \times 3$ identity matrix. Which of the following statements is \textbf{INCORRECT}?

Options

A.
$\text{adj}(A) = I_3 - A$
B.
$B^2 = I_3 \text{ and } |B| = 1$
C.
$|\text{adj}(B) + \text{adj}(A)| = 2$
D.
$\text{adj}(A + B) = A$

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

Set Identity Laws|Idempotent and Involutory Properties|Singular vs Non-Singular Matrices|Properties of Transpose Determinant

Why This Question Is Tricky

Examiner Levers

Negative Thinking

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

Difficulty Boost : +2

Concept Bridge

Cross-chapter synthesis (e.g., using Logarithms in Electrochemistry).

Difficulty Boost : +5

Recursive Logic

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

Difficulty Boost : +5

Common Mistakes

The question tests the synthesis of idempotent matrices, involutory matrices, and the rank-nullity theorem applied to adjugates.

The primary trap lies in Option D: students who correctly derive or verify adj(A) = I_3 - A often falsely assume a symmetric algebraic relation adj(I_3 - A) = A by blindly substituting I_3 - A into the identity.

They fail to recognize that while A has a rank of 2 (allowing a non-zero rank-1 adjugate), I_3 - A has a rank of 1, meaning all its 2x2 minors are zero, forcing adj(I_3 - A) to be the null matrix.

This exposes a critical misunderstanding of how determinant properties and adjugates collapse for singular matrices of varying ranks.

Quick Facts About This Question

Question Information

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