Question

If A and B are two matrices such that rank of A $ = m $ and rank of B $ = n $ then
A. Rank (AB) $ = mn $
B. Rank (AB) $ \geqslant $ rank (A)
C. Rank (AB) $ \geqslant $ rank (B)
D. Rank (AB) $ \leqslant $ min (rank A, rank B)

Answer 1

Hint: Here we are asked to find the rank of the product of the two matrices if the rank of the individual matrix is given. Here we will use the property of the rank of the matrix which are as
Rank of the matrix can be defined as the dimension of the range R(M) of matrix M. so, the rank(M) $ = $ dim(R(M))
Rank can also be defined as dim(V) $ \leqslant $ dim(W) if V is a subset of vector space W.
R(AB) $ \leqslant $ R(A) means the range of matrix AB is less than or equal to the range of matrix A.

Complete step by step solution:
Here given that two matrices A and B with ranks as m and n respectively.
By using the property that the ratio of a matrix M is the dimension of the range of the matrix M.
Also, the range of the matrix M can be written as P(M).
Therefore, by using the property the rank of the matrix M can be re-written as
Rank(M) $ = $ dimR(M)
Now, finding the rank of matrix AB
 $ rank(AB) = \dim (R(AB)){\text{ }}.....{\text{ (1)}} $
Now, the rank of matrix A, $ rank(A) = \dim (R(A))......{\text{ (2)}} $
By property, the range of the product of two matrices A and B is less than or equal to the range of the individual matrix. We can express as –
 $ R(AB) \leqslant R(A),R(AB) \leqslant R(B).......(3) $
Now, for V to be a subset of the vector space W, dimension of V is less than or equal to the dimensions of W.
 $ \therefore \dim (V) \leqslant \dim (W)......(4) $
By using the equations $ (1),(2),(3){\text{ and (4)}} $ we get,
 $ rank(AB) = \dim (R(AB)) \leqslant \dim (R(A)) = rank(A) $
Hence, we can conclude that,
 $ rank(AB) \leqslant rankA $
Similarly, $ rank(B) = \dim (R(B)),R(AB) \leqslant R(B) $
Hence,
 $
  rank(AB) = \dim (R(AB)) \leqslant \dim R(B) = rank(B) \\
   \Rightarrow rank(AB) \leqslant rankB \;
  $
We can observe that,
 $ rank(AB) \leqslant rankA{\text{ and rank}}(AB) \leqslant rank(B) $
Hence, if rank A is less than rank B then rank AB will be less than or equal to the rank A.
Therefore, from the given multiple choices, the option D is the correct answer.
So, the correct answer is “Option D”.

Note: You should know the definitions of the rank of the matrix, range of the matrix and the dimension of the matrix. Know the properties of the rank of the matrix and apply accordingly.