Solving (A+B)^-1: Is A^-1+ B^-1 the Answer?

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Discussion Overview

The discussion revolves around the mathematical expression for the inverse of the sum of two matrices, specifically whether \((A+B)^{-1}\) is equal to \(A^{-1} + B^{-1}\). The scope includes theoretical exploration and mathematical reasoning regarding matrix inverses.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question whether \((A+B)^{-1} = A^{-1} + B^{-1}\) holds true in general.
  • One participant proposes a specific case where \(A = B = I\) to explore the implications of the statement.
  • Another participant provides a detailed derivation showing that \((A+B)^{-1} = A^{-1} - (I + A^{-1}B)^{-1} A^{-1} B A^{-1}\), concluding that the original statement is not true in general.
  • There is a challenge regarding the relationship between \(B^{-1}\) and the derived expression, with participants expressing uncertainty about the conditions under which the equality might hold.
  • One participant emphasizes that the original statement can only be true under specific conditions related to the derived expression for \(B^{-1}\).

Areas of Agreement / Disagreement

Participants do not reach a consensus; multiple competing views remain regarding the validity of the original statement and the conditions under which it might hold true.

Contextual Notes

Assumptions about the invertibility of \(A\), \(B\), and \(A+B\) are critical to the discussion, and the implications of these assumptions are not fully resolved.

roscany
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Does (A+B)^-1= A^-1+ B^-1?

Thanks!
 
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roscany said:
Does (A+B)^-1= A^-1+ B^-1?

Thanks!

Hi roscany,

Suppose we pick A = B = I, what do you get then?
 
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.
 
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.

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)
 
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)

I am saying that in general,
$(A+B)^{-1}=A^{-1}-(I + A^{-1}B)^{-1} A^{-1} B A^{-1}$
Now, the OP's statement that $(A+B)^{-1}=A^{-1}+B^{-1}$ is true if and only if $B^{-1}=-(I + A^{-1}B)^{-1} A^{-1} B A^{-1}$. Since you were able to point out a counter-example, clearly the RHS cannot be simplified to $B^{-1}$.
 

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