My attempt :
$(A^{-1}+B^{-1})(A+B)=I $ . $A^{-1}B + B^{-1}A=-I . $
$| A^{-1}BA|=|-A-B^{-1}A^{2}. | .$
$|B|=|I + B^{-1}A||A|.$ . I am stuck over here
2026-04-03 20:56:03.1775249763
If $A^{-1} + B^{-1} =(A+B)^{-1}$ where A and B are non singular n×n matrices with real entries . Prove that $|A|=|B|$
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$$(A^{-1}+B^{-1})(A+B)=I \\ I+A^{-1}B+B^{-1}A+I=I\\ A^{-1}B+B^{-1}A+I=0 $$
Multiply on the left by $A$ and then by $B$.
$$B+AB^{-1}A+A=0\\ BA^{-1}B+A+B=0$$
This gives $$BA^{-1}B=AB^{-1}A$$
Apply the det and you are done.