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Question #41047

Difficulty: 8/10 | 🟢 Free

The Question

Let $P = \begin{bmatrix} p_1 \\ p_2 \\ p_3 \end{bmatrix}$ be a probability vector such that $p_i \ge 0$ for $i=1,2,3$ and $\sum_{i=1}^3 p_i = 1$ . Consider the skew-symmetric matrix $A = \begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0 \end{bmatrix}$ and the column vector $B = \begin{bmatrix} -1 \\ 2 \\ -1 \end{bmatrix}$ . If $P$ satisfies the system of linear equations $AP = B$ , then the value of $81(p_1^2 + p_2^2 + p_3^2)$ is:

Options

A.
35
B.
45
C.
25
D.
55

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

Axiomatic Probability Constraints|Skew-Symmetric Determinant|System of Linear Equations (Cramer/Inverse)

Why This Question Is Tricky

Common Mistakes

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

The trap lies in the system of equations $AP = B$ .

Because $A$ is an odd-order skew-symmetric matrix, its determinant is strictly zero, making the system either inconsistent or dependent (having infinitely many solutions).

Students who blindly attempt to use Cramer's rule or matrix inversion will fail.

The correct path requires recognizing the linear dependence, expressing the variables parametrically (finding the Null Space translation), and utilizing the fundamental probability constraint $\sum p_i = 1$ as the final independent equation to lock in the unique valid distribution.

Quick Facts About This Question

Question Information

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