Question

If A and B are two invertible matrices of the same order, then adj (AB) is equal to

• A. adj B adj A
• B. $|B||A|B^{-1}{A}^{-1}$
• C. $|B||A|A^{-1}{B}^{-1}$
• D. $|A||B|(AB{)}^{-1}$

Answer 1

Answer: (A), (B), (D)

The correct options are
A

adj B adj A

B

$|B||A|B^{-1}{A}^{-1}$

D

$|A||B|(AB{)}^{-1}$

We have, adj $A=|A|A^{-1}$

adj (AB) =$|AB|(AB{)}^{-1}=|A||B|(AB{)}^{-1}$

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

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

$=adjBadjA$