Question

A is a matrix of order n and k is a constant, then adj(kA) is

• A. $k^{n-1}\mathrm{adj}\left(A\right)$
• B. $k^n\mathrm{adj}\left(A\right)$
• C. $k^{n-2}\mathrm{adj}\left(A\right)$

Answer 1

The correct option is A $k^{n-1}\mathrm{adj}\left(A\right)$

${\left(\mathrm{kA}\right)}^{-1}=\frac{\mathrm{adj}\left(\mathrm{kA}\right)}{\left(}\Rightarrow \mathrm{adj}\left(\mathrm{kA}\right)={\left(\mathrm{kA}\right)}^{-1}\left(\mathrm{kA}\right|=\left({\left(A\right)}^{-1}{k}^{-1}\right)\left(\left(A\right|k^n\right)\mathrm{On}\mathrm{rearranging}\mathrm{the}\mathrm{terms},\mathrm{we}\mathrm{get}-\mathrm{adj}\left(\mathrm{kA}\right)=\left(k^{n-1}\right)\left({\left(A\right)}^{-1}\left(A\right|\right)=\left(k^{n-1}\right)\mathrm{adj}\left(A\right)$