Question

Let A, B , C, D be n$\times$ n matrices, each with non zero determinant and ABCD = I then $B^{-1}=$

• A. $D^{-1}{C}^{-1}{A}^{-1}$
• B. CDA
• C. ABC
• D. Does not exist

Answer 1

The correct option is B CDA

$\left(ABCD\right)=I$

$A^{-1}\left(ABCD\right)D^{-1}=A^{-1}I{D}^{-1}$

$BC=A^{-1}$ $D^{-1}$

$\left(BC\right)C^{-1}$ = $A^{-1}$ $D^{-1}$ $C^{-1}$

$B=A^{-1}$ $D^{-1}$ $C^{-1}$ $\Rightarrow$ $B^{-1}$= $(A^{-1}$ $D^{-1}$ $C^{-1}{)}^{-1}$

So $B^{-1}$ = $(C^{-1}{)}^{-1}$ $(D^{-1}{)}^{-1}$ $(A^{-1}{)}^{-1}$

(using Reversal law)

$=CDA$