Question

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

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

$A$ and $B$ are $2$ invertible matrices of same order. $adj\left(AB\right)=?$

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

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

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