If A and B are two non-singular matrices which commute, then $\left(A (A+B)^{-1} B\right)^{-1}(AB) =$
Answer & explanation
Correct answer: option 1
$\left(A (A+B)^{-1} B\right)^{-1}$
$=B^{-1} (A+B)A^{-1}$ $[∵(ABC)^{-1}=C^{-1}B^{-1}A^{-1}]$
$=B^{-1}AA^{-1}+ B^{-1}BA^{-1}$
$=B^{-1}I+IA^{-1}=B^{-1}+ A^{-1}$
$∴\left(A (A+B)^{-1} B\right)^{-1}(AB) =(B^{-1}+ A^{-1})AB$
$=B^{-1}(AB) +A^{-1}(AB)$
$=B^{-1}(BA) +A^{-1}(AB)$ $[∵AB=BA]$
$=(B^{-1}B)A+(A^{-1}A)B$
$=IA+IB= A + B$.