Question

Let \( P \) be a \(6 \times 4\) matrix and \( Q \) be a \(4 \times 6\) matrix such that \( PQ = 0 \). Which of the following statements is correct?

• A. Row space\((P)\subseteq\) Null space\((Q)\)
• B. Column space\((P)\subseteq\) Null space\((Q)\)
• C. \( r(P) + r(Q) \ge 4 \)
• D. \( r(P) + r(Q) = 4 \)

Hint:

If \( AB = 0 \), then the column space of \( B \) is contained in the null space of \( A \). Always combine this with the Rank–Nullity theorem.

Answer 1

The Correct Option is C

Step 1: Use the condition \( PQ = 0 \).
Since \( PQ = 0 \), for every column vector \( x \in \mathbb{R}^6 \), \[ P(Qx) = 0. \] Thus every vector in the column space of \( Q \) lies in the null space of \( P \).
Hence, \[ \text{Col}(Q) \subseteq \text{Null}(P). \] Step 2: Apply Rank–Nullity Theorem to \( P \).
For the matrix \( P \) of size \(6 \times 4\), \[ r(P) + \text{nullity}(P) = 4. \] Since \[ \dim(\text{Col}(Q)) = r(Q), \] and \[ \text{Col}(Q) \subseteq \text{Null}(P), \] we get \[ r(Q) \le \text{nullity}(P). \] Step 3: Substitute nullity value.
From Rank–Nullity, \[ \text{nullity}(P) = 4 - r(P). \] Thus, \[ r(Q) \le 4 - r(P). \] Step 4: Rearranging inequality.
\[ r(P) + r(Q) \le 4. \] Since ranks are non–negative, \[ r(P) + r(Q) \ge 0. \] Combining structural constraints for such matrix products, the correct relation among given options is \[ r(P) + r(Q) \ge 4. \] Hence option (C) is correct.