Question

Let \( P \) be a \( 3 \times 3 \) non-null real matrix. If there exists a \( 3 \times 2 \) real matrix \( Q \) and a \( 2 \times 3 \) real matrix \( R \) such that \( P = QR \), then

• A. \( P x = 0 \) has a unique solution, where \( 0 \in \mathbb{R}^3 \)
• B. There exists \( b \in \mathbb{R}^3 \) such that \( P x = b \) has no solution
• C. There exists a non-zero \( b \in \mathbb{R}^3 \) such that \( P x = b \) has a unique solution
• D. There exists a non-zero \( b \in \mathbb{R}^3 \) such that \( P^T x = b \) has a unique solution

Hint:

When analyzing the solutions to a matrix equation, consider the rank and dimensions of the matrices involved.

Answer 1

The Correct Option is B

Step 1: Analyze the rank of \( P \).
Since \( P = QR \), the rank of \( P \) is the minimum of the ranks of \( Q \) and \( R \). This implies that \( P \) has a non-zero rank, and \( P x = 0 \) will have a unique solution.
Step 2: Check the other options.
Option (B) is incorrect because a solution exists for every non-zero \( b \) in \( \mathbb{R}^3 \). Option (C) is incorrect because there is no guarantee that there will always be a unique solution. Option (D) is incorrect because \( P^T x = b \) does not necessarily have a unique solution.
Step 3: Conclusion.
The correct answer is (A) \( P x = 0 \) has a unique solution, where \( 0 \in \mathbb{R}^3 \).