If A and B matrices commute then

Options

A. `A^(-1) and B ` also commute
B. ` B^(-1) ` and A also commute
C. `A^(-1) and B^(-1) ` also commute
D. all the above

Answer

To solve the problem, we need to analyze the properties of matrices A and B when they commute.

### Step-by-Step Solution:

1. **Definition of Commutative Matrices**:
- Two matrices A and B are said to commute if \( AB = BA \).

2. **Taking the Inverse**:
- If A and B commute, we can take the inverse of the product \( AB \). The property of inverses states that \( (AB)^{-1} = B^{-1}A^{-1} \).

3. **Using the Commutative Property**:
- Since \( AB = BA \), we can also express the inverse as \( (BA)^{-1} = A^{-1}B^{-1} \).

4. **Equating the Two Inverses**:
- From the above, we have:
\[ (AB)^{-1} = B^{-1}A^{-1} \quad \text{and} \quad (BA)^{-1} = A^{-1}B^{-1} \]
- Since \( AB = BA \), we can conclude that:
\[ B^{-1}A^{-1} = A^{-1}B^{-1} \]

5. **Conclusion about Inverses**:
- The equality \( B^{-1}A^{-1} = A^{-1}B^{-1} \) shows that the inverses of A and B also commute. Thus, we can say that if A and B commute, then their inverses \( A^{-1} \) and \( B^{-1} \) also commute.

### Final Statement:
If matrices A and B commute, then their inverses \( A^{-1} \) and \( B^{-1} \) also commute.