Question

Assertion :If the matrices A, B, (A + B) are nonsingular, then $\left[A\right(A+B{)}^{-1}B{]}^{-1}=B^{-1}+A^{-1}$ Reason: $\left[A\right(A+B{)}^{-1}B{]}^{-1}=\left[A\right(A^{-1}+B^{-1})B{]}^{-1}=[(I+A{B}^{-1})B{]}^{-1}=\left[\right(B+A{B}^{-1}B){]}^{-1}=[(B+AI){]}^{-1}=\left[\right(B+1){]}^{-1}=B^{-1}+A^{-1}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

Assertion

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

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

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

$\Rightarrow {[A{B}^{-1}B+IB]}^{-1}$

$\Rightarrow {[A+B]}^{-1}=B^{-1}+A^{-1}$

True

Reason:False

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

But given ${(B+AI)}^{-1}={(B+I)}^{-1}$

Option C