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Question #40368

Difficulty: 8/10 | 🟢 Free

The Question

For any two $3 \times 3$ matrices $X$ and $Y$ , define the operation $X \oplus Y = X + Y - XY$ . Let $A$ be a symmetric $3 \times 3$ real matrix such that $A \oplus A = A$ and $\text{Tr}(A) = 2$ . If $B = I - 2A$ , then the value of $\det(\text{adj}(B^T B + 2B + 3I))$ is

Options

A.
576
B.
64
C.
24
D.
46656

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

Common Mistakes

This question maps the set union identity ( $A \cup A = A$ ) to matrix algebra via the custom operation $A \oplus A = A$ , which simplifies to the idempotent property $A^2 = A$ .

The primary trap tests the student's handling of singular matrices: students often incorrectly deduce from $A(I-A)=0$ that $A=I$ or $A=0$ , ignoring that matrices can be zero divisors.

If a student assumes $A=I$ , they will incorrectly calculate the final determinant as 64.

Furthermore, the question tests the involutory property ( $B^2=I$ ) and the adjugate determinant relation $\det(\text{adj}(M)) = \det(M)^{n-1}$ , trapping students who forget to square the determinant (yielding 24).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sets - Class 11
Concept	Set Identity Laws
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free