Question

Prove that : (a+b)^-1 (a^-1+b^-1)= 1/ab​

Answer 1

Answer:

Multiply both sides by A+B on the right, to get

I=A−1(A+B)+B−1(A+B)=I+A−1B+B−1A+I

and so

A−1B=−B−1A−I

Multiplying by B on the left, we get BA−1B=−A−B .

If we multiply by A+B on the left we get

I=(A+B)A−1+(A+B)B−1=I+BA−1+AB−1+I

and so

AB−1=−BA−1−I

Multiplying by A on the right, we obtain

AB−1A=−B−A

Since −B−A=−A−B , we conclude

AB−1A=BA−1B=−(A+B)

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