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Question #51348

Difficulty: 8/10 | 🟢 Free

The Question

Let $A$ and $B$ be $3 \times 3$ real matrices satisfying the idempotent property $X^2 = X$ (analogous to the set identity $S \cup S = S$ ). Suppose $AB = BA = O$ , $\det(I - 2A) = -1$ , and $\det(I - 2B) = 1$ . If $B$ is a non-null matrix, find the value of $\log_2(\det(A - B + 3I))$ .

Options

A.
3
B.
4
C.
5
D.
6

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

Concept Bridge

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

Difficulty Boost : +5

Common Mistakes

This question tests the synthesis of idempotent matrix properties and simultaneous diagonalizability, mapping set identity laws ( $X \cup X = X$ ) to matrix algebra ( $X^2 = X$ ).

The primary trap lies in the determinant conditions: students often incorrectly assume that if $A^2 = A$ , then $\det(A) = 1$ (ignoring the singular case $\det(A) = 0$ ), which would falsely imply $A = I$ and $B = O$ .

Furthermore, it tests the involutory nature of $I - 2A$ .

By forcing students to deduce the exact eigenvalue spectrums ( $\lambda_A=\{1,0,0\}$ and $\lambda_B=\{0,1,1\}$ ) and use $AB=O$ to prove $A+B=I$ , it punishes those who attempt to blindly expand the determinant without analyzing the underlying eigenbasis.

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	170 sec
Access	Free