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Question #41073

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all $2 \times 2$ matrices with entries chosen independently and uniformly from the set $\{-1, 0, 1\}$ . A matrix $A$ is chosen at random from $S$ . Given that $A$ is non-singular, what is the probability that $\left[ \frac{\det(A)}{2} \right] = 0$ , where $[x]$ denotes the greatest integer less than or equal to $x$ ?

Options

A.
$\frac{5}{12}$
B.
$\frac{5}{6}$
C.
$\frac{53}{81}$
D.
$\frac{73}{81}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Conditional Probability Foundations|Singular vs. Non-Singular Threshold|Greatest Integer Function (GIF)

Why This Question Is Tricky

Common Mistakes

This question tests conditional probability combined with the Greatest Integer Function (GIF) acting as a threshold on a discrete determinant space.

The primary trap lies in the evaluation of the GIF for negative values: students frequently assume $[-0.5] = 0$ instead of $-1$ , leading them to incorrectly include $\det(A) = -1$ in the favorable outcomes (yielding the trap answer 5/6).

Additionally, it tests the ability to correctly redefine the sample space by excluding singular matrices ( $\det(A) = 0$ ), penalizing those who calculate the absolute probability instead of the conditional one.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 11
Concept	Conditional Probability Foundations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free