Discussion
Let A be a 6 × 7 matrix of rank 6 and B be a 7 × 5 matrix of rank 4, then the rank of AB is :
Answer: D
The rank of AB cannot be determined exactly, but is at most 4.
Upper bound: \(\text{rank}(AB) \le \min(\text{rank } A, \text{rank } B) = \min(6, 4) = 4\).
Lower bound, by Sylvester's inequality with inner dimension 7:
\[ \begin{aligned} \text{rank}(AB) &\ge 6 + 4 - 7 \\ &= 3 \end{aligned} \]
Both values occur. A has a one-dimensional null space; if that null space lies inside the column space of B, AB loses one dimension and has rank 3, otherwise it keeps rank 4. Since the data do not decide this, the rank is 3 or 4, so the exact values 4, 5 and 6 are all wrong as certainties (5 and 6 are impossible).
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