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Question #41829

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all $3 \times 3$ matrices with entries from the set $\{0, 1\}$ . A matrix $A$ is chosen at random from $S$ . What is the probability that $\det(A) > 0$ ?

Options

A.
$\frac{87}{512}$
B.
$\frac{21}{128}$
C.
$\frac{21}{64}$
D.
$\frac{87}{256}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Complementary Counting|Linear Dependency of Rows|Row/Column Exchange Parity

Why This Question Is Tricky

Common Mistakes

This question tests complementary counting and linear dependency over finite fields versus real numbers, specifically focusing on the row exchange parity trap.

Students often calculate the number of non-singular matrices over $\mathbb{Z}_2$ (168) and correctly apply the parity argument (dividing by 2) to find matrices with positive determinants, yielding 84.

The trap is assuming that $\mathbb{Z}_2$ non-singularity covers ALL real non-singular matrices.

They miss the matrices with determinant $\pm 2$ , which are singular over $\mathbb{Z}_2$ but non-singular over $\mathbb{R}$ .

Furthermore, they must correctly apply the row-exchange parity (Leibniz formula sign flips) to these extra matrices to find that exactly 3 of them have a determinant of +2, leading to the correct numerator of 87.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Complementary Counting
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free