Q. If $A$, $B$ and $(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1}))$ are non-singular matrices of same order, then the inverse of $A\left(\operatorname{adj}(A^{-1})+\operatorname{adj}(B^{-1})\right)^{-1}B$, is equal to
A) $AB^{-1}+A^{-1}B$
B) $\operatorname{adj}(B^{-1})+\operatorname{adj}(A^{-1})$
C) $\frac{AB^{-1}}{|A|}+\frac{BA^{-1}}{|B|}$
D) $\frac1{|AB|}\left(\operatorname{adj}(B)+\operatorname{adj}(A)\right)$
Explanation
[Solution by Dr. Abubacker] $\operatorname{adj}(A^{-1})=\frac A{|A|}$ and $\operatorname{adj}(B^{-1})=\frac B{|B|}$. The inverse of the product is $B^{-1}\left(\frac A{|A|}+\frac B{|B|}\right)A^{-1}=\frac{B^{-1}}{|A|}+\frac{A^{-1}}{|B|}=\frac{|B|B^{-1}+|A|A^{-1}}{|A||B|}=\frac1{|AB|}\left(\operatorname{adj}B+\operatorname{adj}A\right)$. Answer (4).
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.