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Question #41078

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the set of all $4 \times 4$ matrices with entries from the set $\{0, 1\}$ . A matrix $M$ is chosen at random from $S$ such that all four rows of $M$ are distinct and non-zero. What is the probability that the determinant of $M$ is an even integer?

Options

A.
$\frac{5}{13}$
B.
$\frac{8}{13}$
C.
$\frac{7}{13}$
D.
$\frac{6}{13}$

Check Your Answer

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

Multiplication Theorem|Linear Independence vs Dependence|Linear Dependency of Rows

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of combinatorial probability and linear algebra.

The trap lies in evaluating the determinant constraint.

Students often attempt to manually count matrices with determinant 0 using inclusion-exclusion or cofactor expansion, which is computationally impossible in a timed setting.

The elegant leap is recognizing that the parity of the determinant over $\mathbb{Z}$ is isomorphic to the determinant over the finite field $\mathbb{Z}_2$ .

This transforms the problem into finding the probability of linear dependence of rows over $\mathbb{Z}_2$ , solvable via the geometric series of available independent choices $(q^n - 1)(q^n - q)\dots$ rather than brute-force counting.

Quick Facts About This Question

Question Information

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