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

  1. $D^{-1}C^{-1}A^{-1}$
  2. $CDA$
  3. $ADC$
  4. Does not necessarily exist.
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(B)
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2 Answers

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Given,   $ABCD=I$   and all the matrices are invertible (as their determinant is non-zero given)

Pre-multiplying on both sides of the equation with  $A^{-1}$,   $A^{-1}ABCD=A^{-1}I$   $\Rightarrow$   $BCD=A^{-1}$

Pre-multiplying on both sides of the equation with  $B^{-1}$,   $B^{-1}BCD=B^{-1}A^{-1}$   $\Rightarrow$   $CD=B^{-1}A^{-1}$

Post-multiplying on both sides of the equation with  $A$,   $CDA=B^{-1}A^{-1}A$   $\Rightarrow$    $CDA=B^{-1}$

Option B is correct.
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