Question

If $A$ and $B$ are non singular matrices which commute then $A\left(A\right(A+B{)}^{-1}B{)}^{-1}B$$=$

• A. $A+B$
• B. $A^{-1}+B^{-1}$
• C. $A^{-1}+B$
• D. $A-B$

Answer 1

Answer: (A)

$A+B$

Now,

$A\left(A\right(A+B{)}^{-1}B{)}^{-1}B$

$=A\left(A\right(A^{-1}+B^{-1})B{)}^{-1}B$

$=A\left(\right(A{A}^{-1}+A{B}^{-1})B{)}^{-1}B$

$=A\left(\right(I+A{B}^{-1})B{)}^{-1}B$ [ Since $A{A}^{-1}=I$]

$=A\left(\right(B+A{B}^{-1}B){)}^{-1}B$ [ Since $B^{-1}B=I$]

$=A\left(\right(B+A){)}^{-1}B$

$=A\left(\right(B^{-1}+A^{-1}\left)\right)B$

$=(A{B}^{-1}+A{A}^{-1})B$

$=(A{B}^{-1}+I)B$

$=(A{B}^{-1}B+B)$

$=A+B$.