We are tasked with proving the equality:

$\left(A^{-1} + B^{-1}\right)^{-1} = A\left(B + A\right)^{-1}B$

Step-by-step Proof:

Given:

• $A$ and $B$ are invertible square matrices.
• $A + B$ is also invertible.

We need to show that:

$\left(A^{-1} + B^{-1}\right)^{-1} = A\left(B + A\right)^{-1}B$

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Step 1: Start with the left-hand side

We begin with the left-hand side of the equation, which is $\left(A^{-1} + B^{-1}\right)^{-1}$.

We can manipulate $A^{-1} + B^{-1}$ as follows:

$A^{-1} + B^{-1} = A^{-1}(A + B)B^{-1}$

This follows from the identity:

$A^{-1} + B^{-1} = A^{-1} \cdot I + I \cdot B^{-1} = A^{-1}(A + B)B^{-1}$

Now, the left-hand side becomes:

$\left(A^{-1} + B^{-1}\right)^{-1} = \left(A^{-1}(A + B)B^{-1}\right)^{-1}$

Step 2: Inverse of a product of matrices

The inverse of a product of matrices $X \cdot Y$ is given by:

$(XY)^{-1} = Y^{-1} X^{-1}$

So, applying this to our expression:

$\left(A^{-1}(A + B)B^{-1}\right)^{-1} = B \cdot (A + B)^{-1} \cdot A$

This simplifies to:

$= A \cdot (A + B)^{-1} \cdot B$

Thus, we have shown that the left-hand side is equal to $A \cdot (A + B)^{-1} \cdot B$, which is exactly the right-hand side of the equation.

Conclusion

We have successfully shown that:

$\left(A^{-1} + B^{-1}\right)^{-1} = A\left(B + A\right)^{-1}B$

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Further Clarification

Would you like any further clarification on specific steps or a deeper explanation of any part of the proof?

Related Questions:

1. How does the inverse of a matrix behave under matrix addition and multiplication?
2. What is the condition under which $(A + B)^{-1}$ exists for two invertible matrices $A$ and $B$?
3. How do you prove the identity $(AB)^{-1} = B^{-1}A^{-1}$?
4. What happens when one of the matrices, say $A$ or $B$, is not invertible in this proof?
5. How can this result be generalized to more than two matrices?

Tip: Always remember that matrix inverses and products have specific properties that allow us to manipulate complex expressions—like how the inverse of a product is the product of the inverses in reverse order!