Question

Let \( m, n \in \mathbb{N}, m < n, P \in M_{n \times m}(\mathbb{R}), Q \in M_{m \times n}(\mathbb{R}) \). Which of the following is (are) NOT possible? 
 

• A. \( \text{rank}(PQ) = n \)
• B. \( \text{rank}(QP) = m \)
• C. \( \text{rank}(PQ) = m \)
• D. \( \text{rank}(QP) = \left\lceil\dfrac{m+n}{2}\right\rceil \) the smallest integer larger than or equal to \(\dfrac{m+n}{2}\)

Hint:

When dealing with matrix ranks, remember that the rank of a product is limited by the ranks of the individual matrices involved.

Answer 1

The Correct Option is A, D

Key facts

\(\operatorname{rank}(PQ) \le \min(\operatorname{rank} P,\operatorname{rank} Q)\le m\)

\(\operatorname{rank}(QP) \le m\) since (QP) is \(m\times m\)

Now check options:

(A) \(\operatorname{rank}(PQ)=n\)
Impossible since \(\operatorname{rank}(PQ)\le m<n\).

(B) \(\operatorname{rank}(QP)=m\)
Possible (take (QP) full rank).

(C) \(\operatorname{rank}(PQ)=m\)
Possible (maximum attainable rank).

(D) \(\operatorname{rank}(QP)=\left\lceil\frac{m+n}{2}\right\rceil\)
Since (n>m), \(\left\lceil\frac{m+n}{2}\right\rceil>m\), but \(\operatorname{rank}(QP)\le m\). Impossible.

Correct answer (NOT possible): A and D