Question

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

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

Answer 1

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

The correct options are
A $adj\left(B\right)adj\left(A\right)$
B $\left|B\right|\left|A\right|B^{-1}{A}^{-1}$
D $\left|A\right|\left|B\right|(AB{)}^{-1}$

Given A and B are invertible matrices of some order

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

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

Now,

adj$\left(AB\right)=\left|AB\right|({AB)}^{\_1}$

adj$\left(AB\right)=\left|A\right|\left|B\right|({AB)}^{\_1}$ Option D is correct

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

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

adj$\left(AB\right)=\left(adjB\right)\left(adjA\right)$ Option A is correct

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

$=\left|B\right|\left|A\right|B^{-1}{A}^{-1}$ Option B is correct