Let A and B be two `3xx3` invertible matrices . If A + B = AB then

Options

A. `A^(-1)+B^(-1)=O`
B. `A^(-1)+B^(-1)= B^(-1)A^(-1)`
C. `I-A^(-1)` is invertible
D. `B^(-1)+I` is invertible

Answer

To solve the problem where \( A + B = AB \) for two invertible matrices \( A \) and \( B \), we can follow these steps:

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ A + B = AB \]

2. **Rearrange the equation:**
\[ AB - A - B = 0 \]

3. **Add \( I \) (the identity matrix) to both sides:**
\[ AB - A - B + I = I \]

4. **Factor the left-hand side:**
\[ (A - I)(B - I) = I \]
This implies that \( A - I \) and \( B - I \) are inverses of each other.

5. **Taking the inverse of both sides:**
Since \( (A - I)(B - I) = I \), we can express \( B - I \) in terms of \( A - I \):
\[ B - I = (A - I)^{-1} \]

6. **Rearranging gives us:**
\[ B = (A - I)^{-1} + I \]

7. **Now, let's find \( A^{-1} + B^{-1} \):**
Using the relationship we derived:
\[ B^{-1} = (A - I) + I = A \]
Therefore:
\[ A^{-1} + B^{-1} = A^{-1} + (A - I)^{-1} \]

8. **Now we can analyze the options provided in the question.**
We need to check the conditions for invertibility and the relationships derived.

### Conclusion:
From the steps above, we can conclude that \( A^{-1} + B^{-1} \) does not equal zero, but it leads us to the conclusion that \( B^{-1} = I - A^{-1} \) is an invertible matrix.