Question

If A and B are two matrices such that rank of $A=m$ and rank of $B=n$ then

• A. $rank\left(AB\right)=mn$
• B. $rank\left(AB\right)\ge rank\left(A\right)$
• C. $rank\left(AB\right)\ge rank\left(B\right)$
• D. $rank\left(AB\right)\le min(rankA,rankB)$

Answer 1

The correct option is D $rank\left(AB\right)\le min(rankA,rankB)$

The matrix AB is actually a matrix that consist the linear combination of A with B the multipliers.

Suppose if B is singular, then when B, being the multipliers of A, will naturally obtain another singular matrix of AB. Similarly, if B is non-singular, then AB will be non-singular. Therefore, the $rank\left(AB\right)<rank\left(B\right)$.

Then now if A is singular, then clearly, no matter what B is, the $rank\left(AB\right)<rank\left(A\right)$. The$rank\left(AB\right)$ is immediately capped by the rank of A unless the the rank of B is even smaller.

Put these two ideas together, the rank of AB must have been capped the rank of A or B, which ever is smaller. Therefore,$rank\left(AB\right)<min\left(rank\right(A),rank(B\left)\right)$.