GATE | Sudo GATE 2020 Mock III (24 January 2019) | Question 14

If A as m×n matrix with rank n, and B as n×p matrix with rank p. Then the rank of matrix C=AB is __________ .
(A) m
(B) n
(C) p
(D) None of these.

Answer: (C)
Explanation: The conditions from the hypothesis imply that m≥n≥p. One knows that,

rank(AB) ≤ min(m,p) = p 

On the other side, from Sylvester Rank Inequality we get,

p = n+p−n ≤ rank(AB) 

So,

rank(AB) = p

Option (C) is correct.
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