Question

If $A$ is invertible matrix and $B$ is any matrix, then

• A. Rank $(AB) =$ Rank $(A)$
• B. Rank $(AB) =$ Rank $(B)$
• C. Rank $(AB) >$ Rank $(A)$
• D. Rank $(AB) >$ Rank $(B)$

Answer 1

Answer: (B)

Since, $A$ is invertible $\Rightarrow A^{-1}$ exists.
Now, Rank $(B)=\operatorname{Rank}\left(A^{-1}(A B)\right)$
$\leq \operatorname{Rank}(A B) [\because Rank(P Q) \leq \operatorname{Rank} Q]$
But, Rank $(A B) \leq \operatorname{Rank}(B)$
$\therefore $ Rank $(A B)=\operatorname{Rank}(B)$