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Question #41245

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all $3 \times 3$ matrices with entries from $\{-1, 1\}$ . A matrix $A$ is chosen at random from $S$ . Let $V$ be the volume of the parallelepiped formed by the column vectors of $A$ . Let $E$ be the event that the sum of the roots of the characteristic equation of $A$ is $3$ . Find the conditional probability $P(V = 0 \mid E)$ .

Options

A.
$\frac{5}{8}$
B.
$\frac{1}{2}$
C.
$\frac{3}{8}$
D.
$\frac{5}{16}$

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

Characteristic Equation|Determinant as Geometric Area/Volume|Equiprobability Fallacy

Why This Question Is Tricky

Common Mistakes

The question tests the intersection of linear algebra (characteristic equation, determinant as volume) and combinatorics.

The 'Equiprobability Fallacy' trap is laid by asking for the probability of V=0.

Students often assume that the determinant values are uniformly distributed or fail to recognize the hidden algebraic dependency (the product of the expansion terms equals -1) among the terms of the determinant expansion.

By mapping the determinant terms to an affine transformation over a finite field, the problem forces students to rigorously evaluate the constrained probability space rather than relying on intuitive but flawed uniform counting.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Matrices - Class 12
Concept	Characteristic Equation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free