Question

Let \(m, n \in \mathbb{N}\) such that \(m<n\) and \(P_{m \times n}(\mathbb{R})\) and \(Q_{n \times m}(\mathbb{R})\) are matrices over real numbers and let \(\rho(V)\) denotes the rank of the matrix V. Then, which of the following are NOT possible.
A. \( \rho(PQ) = n \)
B. \( \rho(QP) = m \)
C. \( \rho(PQ) = m \)
D. \( \rho(QP) = \lfloor(m+n)/2\rfloor \), where \(\lfloor \rfloor\) is the greatest integer function

• A. A and D only
• B. B and C only
• C. A, C and D only
• D. A, B and C only

Hint:

Remember this fundamental rule: the rank of a product of matrices, \(AB\), can never be greater than the rank of either \(A\) or \(B\). This simple rule is often sufficient to quickly eliminate impossible scenarios in rank-related problems.

Answer 1

The Correct Option is A

Step 1: Understanding the Problem: This question concerns the rank of a product of two non-square matrices. We'll utilize fundamental properties of matrix rank, focusing on its relationship to matrix dimensions and products.

Step 2: Key Concept: The core property for the rank of a matrix product \( AB \) is:

\[ \rho(AB) \le \min(\rho(A), \rho(B)) \]

Also, a matrix's rank cannot exceed its number of rows or columns. 
For \(P_{m \times n}\), \( \rho(P) \le \min(m, n) = m \) (since \(m < n\)). 
For \(Q_{n \times m}\), \( \rho(Q) \le \min(n, m) = m \). 

Step 3: Analysis of Statements: 
Let's evaluate each statement: 
A. \( \rho(PQ) = n \): 
The product \(PQ\) has dimensions \( m \times m \). Its maximum rank is \(m\). Given \( m < n \), \( \rho(PQ) \) cannot be \(n\). 
Also, \( \rho(PQ) \le \min(\rho(P), \rho(Q)) \le \min(m, m) = m \). Since \(m < n\), \( \rho(PQ) \) must be less than \(n\). Therefore, \( \rho(PQ) = n \) is impossible. 
B. \( \rho(QP) = m \): 
The matrix product \(QP\) has dimensions \( n \times n \). \( \rho(QP) \le \min(\rho(Q), \rho(P)) \le m \). Thus, the rank is at most \(m\). Matrices \(P\) and \(Q\) can be constructed such that \( \rho(P) = \rho(Q) = m \) and \( \rho(QP) = m \). This statement is possible. 
C. \( \rho(PQ) = m \): 
\(PQ\) is an \(m \times m\) matrix. Its rank can be at most \(m\). It's possible to construct \(P\) and \(Q\) such that \( \rho(P) = \rho(Q) = m \) and \( \rho(PQ) = m \). This statement is possible. 
D. \( \rho(QP) = \lfloor (m+n)/2 \rfloor \): 
We know \( \rho(QP) \le m \). For this to hold, \( \lfloor (m+n)/2 \rfloor \le m \). Since \( m < n \), we can find cases where this is violated. For example, let \( m = 2 \) and \( n = 5 \). Then \( \rho(QP) = \lfloor (2+5)/2 \rfloor = 3 \), but \( \rho(QP) \le m = 2 \). This is a contradiction. Therefore, this statement is generally impossible. 
Step 4: Conclusion: 
Statements A and D describe scenarios that are not generally feasible. Thus, the correct option is (A).