roscany
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Does (A+B)^-1= A^-1+ B^-1?
Thanks!
Thanks!
roscany said:Does (A+B)^-1= A^-1+ B^-1?
Thanks!
Rido12 said:Assume $A$, $B$, and $A+B$ are invertible, otherwise the inverse would not exist. Let $(A+B)^{-1}=A^{-1}+X$, where $X$ is to be determined.
$(A+B)^{-1}=A^{-1}+X$
Multiply both sides by $A+B$:
$I=(A^{-1}+X)(A+B)$
$I=A^{-1}A+A^{-1}B+XA+XB$
$I=A^{-1}B+X(A+B)$
$X(A+B)=-A^{-1}B$
$X=(-A^{-1}B)(A+B)^{-1}$
Recall from above:
$X=(-A^{-1}B)(A^{-1}+X)$
Isolating for $X$...
$X= - (I + A^{-1}B)^{-1} A^{-1} B A^{-1}$
$(A+B)^{-1}=A^{-1}-(I + A^{-1}B)^{-1} A^{-1} B A^{-1}$
Therefore, the original statement is not true in general.
I like Serena said:Wait. (Wait)
Are you saying that $B^{-1}$ does not have to be equal to $-(I + A^{-1}B)^{-1} A^{-1} B A^{-1}$?
Why would that be the case? (Wondering)