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Question #41070

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{-2, -1, 0, 1, 2\}$ . A $2 \times 2$ matrix $A$ is formed by choosing its entries randomly with replacement from $S$ . What is the probability that $A^3 \neq O$ , where $O$ is the $2 \times 2$ null matrix?

Options

A.
$\frac{608}{625}$
B.
$\frac{616}{625}$
C.
$\frac{624}{625}$
D.
$\frac{598}{625}$

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

Complementary Counting|Sample Space Partitioning|Nilpotent Matrix Logic

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of nilpotent matrices and complementary counting.

The primary trap is the assumption that $A^3 = O \implies A = O$ , leading to the incorrect probability of $624/625$ .

Furthermore, students often fail to realize that for a $2 \times 2$ matrix, the index of nilpotency cannot exceed 2, meaning $A^3 = O \iff A^2 = O$ .

By partitioning the sample space based on the trace and determinant conditions ( $\text{Tr}(A)=0, \det(A)=0$ ), and using complementary counting, we avoid the tedious direct calculation of $A^3 \neq O$ and prevent double-counting errors in the matrix space distribution.

Quick Facts About This Question

Question Information

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