If the matrix `A` and `B` are of `3xx3` and `(I-AB)` is invertible, then which of the following statement is/are correct ?

Options

A. `I-BA` is not invertible
B. `I-BA` is invertible
C. `I-BA` has for its inverse `I+B(I-AB)^(-1)A`
D. `I-BA` has for its inverse `I+A(I-BA)^(-1)B`

Explanation

To solve the problem, we need to analyze the given statements based on the condition that the matrix \( I - AB \) is invertible. We will evaluate each statement one by one.

### Given:
- Matrices \( A \) and \( B \) are \( 3 \times 3 \).
- \( I - AB \) is invertible.

### Statements to evaluate:
1. \( I - BA \) is not invertible.
2. \( I - BA \) is invertible.
3. \( I - BA \) has the inverse \( I + B(I - AB)^{-1}A \).
4. \( I - BA \) has the inverse \( I + A(I - BA)^{-1}B \).

### Step-by-step Solution:

**Step 1: Understanding the Invertibility Condition**
- Since \( I - AB \) is invertible, it implies that \( \det(I - AB) \neq 0 \).

**Step 2: Analyzing \( I - BA \)**
- We know from properties of matrices that if \( I - AB \) is invertible, then \( I - BA \) is also invertible. This is a result of the relation between the determinants of \( AB \) and \( BA \) for square matrices of the same size.

**Step 3: Evaluating the Statements**
- **Statement 1:** \( I - BA \) is not invertible.
- This statement is **false** because we established that \( I - BA \) is invertible if \( I - AB \) is invertible.

- **Statement 2:** \( I - BA \) is invertible.
- This statement is **true** based on our previous analysis.

- **Statement 3:** \( I - BA \) has the inverse \( I + B(I - AB)^{-1}A \).
- To verify this, we can multiply \( I - BA \) by \( I + B(I - AB)^{-1}A \) and check if it equals the identity matrix \( I \).
- After performing the multiplication and simplifying, we find that this statement holds true.

- **Statement 4:** \( I - BA \) has the inverse \( I + A(I - BA)^{-1}B \).
- This statement is not valid as it does not follow from the properties of the matrices involved.

### Conclusion:
- The correct statements are:
- **Statement 2**: \( I - BA \) is invertible.
- **Statement 3**: \( I - BA \) has the inverse \( I + B(I - AB)^{-1}A \).

### Final Answer:
- The correct options are **2 and 3**.