    Question

# Assertion :If the matrices A, B, (A + B) are nonsingular, then [A(A+B)−1B]−1=B−1+A−1 Reason: [A(A+B)−1B]−1=[A(A−1+B−1)B]−1=[(I+AB−1)B]−1=[(B+AB−1B)]−1=[(B+AI)]−1=[(B+1)]−1=B−1+A−1

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

## The correct option is C Assertion is correct but Reason is incorrectAssertion[A(A+B)−1B]−1⇒[A(B−1+A−1)B]−1⇒[(AB−1+AA−1)B]−1⇒[AB−1B+IB]−1⇒[A+B]−1=B−1+A−1TrueReason:False ∵(B+AI)−1=(B+A)−1=B−1+A−1But given (B+AI)−1=(B+I)−1Option C  Suggest Corrections  0      Related Videos   Adjoint and Inverse of a Matrix
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