If A and B are two square matrices such that AB=I, then which of the following is not true?

Options

A. BA=I
B. `A^(-1)=B`
C. `B^(-1)=A`
D. `A^(2)=B`

Explanation

To solve the problem, we need to analyze the given condition that \( AB = I \) where \( A \) and \( B \) are square matrices. We need to determine which of the provided statements is not true based on this condition.

### Step-by-Step Solution:

1. **Given Condition**: We start with the equation \( AB = I \), where \( I \) is the identity matrix.

2. **Multiply by \( A^{-1} \)**: We can multiply both sides of the equation \( AB = I \) by \( A^{-1} \) (the inverse of matrix \( A \)):
\[ A^{-1}(AB) = A^{-1}I \]
This simplifies to:
\[ (A^{-1}A)B = A^{-1} \]
Since \( A^{-1}A = I \), we have:
\[ IB = A^{-1} \implies B = A^{-1} \]
This shows that \( B \) is the inverse of \( A \).

3. **Multiply by \( B^{-1} \)**: Next, we can multiply both sides of the original equation \( AB = I \) by \( B^{-1} \):
\[ A(BB^{-1}) = I(B^{-1}) \]
This simplifies to:
\[ A \cdot I = B^{-1} \]
Thus, we have:
\[ A = B^{-1} \]
This shows that \( A \) is the inverse of \( B \).

4. **Check for \( BA = I \)**: Now, we can check if \( BA = I \):
\[ BA = B(A) = B(A^{-1}) = I \]
Since we have already established that \( B = A^{-1} \), we can conclude:
\[ BA = I \]
This means \( BA = I \) is also true.

5. **Conclusion**: Since \( B = A^{-1} \) and \( A = B^{-1} \), all derived equations are valid. However, we need to identify which statement is not true based on the options provided.

### Final Answer:
The statement that is not true among the options provided is the one that contradicts our findings.