If a matrix A has inverses B and C, then which one of the following is correct ?

Options

A. B may not be equal to C
B. B should be equal to C
C. B and C should be unit matrices
D. None of the above

Explanation

To solve the question "If a matrix A has inverses B and C, then which one of the following is correct?", we can use the properties of matrix inverses.

### Step-by-Step Solution:

1. **Understanding Matrix Inverses**:
- If \( A \) is a square matrix and has an inverse, it means there exists a matrix \( B \) such that \( AB = I \) and \( BA = I \), where \( I \) is the identity matrix.
- Similarly, if \( C \) is also an inverse of \( A \), then \( AC = I \) and \( CA = I \).

2. **Equating the Inverses**:
- Since both \( B \) and \( C \) are inverses of \( A \), we can set up the following equations:
\[ AB = I \quad \text{and} \quad AC = I \]

3. **Multiplying by Matrix A**:
- We can multiply both sides of the equation \( AB = I \) by \( C \) from the right:
\[ (AB)C = IC \implies A(BC) = C \]
- Similarly, we can multiply both sides of the equation \( AC = I \) by \( B \) from the left:
\[ B(AC) = BI \implies (BA)C = B \]

4. **Using the Identity Property**:
- Since \( AB = I \), we can replace \( A \) with \( B^{-1} \) in the equations derived:
\[ A = B^{-1} \quad \text{and} \quad A = C^{-1} \]
- This implies that \( B = C \).

5. **Conclusion**:
- Therefore, if a matrix \( A \) has inverses \( B \) and \( C \), it follows that \( B = C \).
- The correct statement is that the two inverses are equal.

### Final Answer:
If a matrix \( A \) has inverses \( B \) and \( C \), then \( B = C \).