Question

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

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

Answer 1

Answer: (B)

$Rank\left(AB\right)=Rank\left(B\right)$

Since, $A$ is invertible $\Rightarrow A^{-1}$ exists.
Now, $Rank\left(B\right)=Rank\left(A^{-1}\right(AB\left)\right)$
$\le Rank\left(AB\right)[\because Rank(PQ)\le RankQ]$
But, $Rank\left(AB\right)\le Rank\left(B\right)$
$\therefore Rank\left(AB\right)=Rank\left(B\right)$.