Question

If A is invertible matrix and B is another 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)

rank (AB)$=$ rank(B)

$A$ is an invertible matrix
i.e $\left|A\right|\ne 0$ so, $A^{-1}$ exist

Note : $rank\left(AB\right)<rankA$ and $rank\left(AB\right)\le rankB$

$B$ = $A^{-1}\left(AB\right)$

$rank\left(B\right)=rank\left(A^{-1}\right(AB\left)\right)$
$\Rightarrow rank\left(B\right)\le rank\left(AB\right)$
but $rank\left(AB\right)\le rankB$
$\therefore rank\left(AB\right)=rank\left(B\right)$