NIELIT 2016 DEC Scientist B (CS) - Section B: 21

Question

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

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

Comments

(B)

Answer 1

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.

Answer 2

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