prove that 
𝐴
−
1
A 
−1
  (the inverse of matrix 
𝐴
A) exists if and only if 
𝐴
−
𝑇
A 
−T
  (the inverse of the transpose of 
𝐴
A, also known as the transpose of 
𝐴
−
1
A 
−1
 ) exists.

To prove that \( A^{-1} \) (the inverse of matrix \( A \)) exists if and only if \( A^{-T} \) (the inverse of the transpose of \( A \), also known as the transpose of \( A^{-1} \)) exists, we can break the proof into two parts: the "if" part and the "only if" part.

### Part 1: **If \( A^{-1} \) exists, then \( A^{-T} \) exists.**

1. **Given**: \( A^{-1} \) exists.  
   - This means there is a matrix \( A^{-1} \) such that:
     \[
     A A^{-1} = A^{-1} A = I
     \]
     where \( I \) is the identity matrix.

2. **Transpose both sides**:  
   Since transposing a matrix product reverses the order of multiplication and transposes each matrix:
   \[
   (A A^{-1})^T = I^T
   \]
   \[
   (A^{-1})^T A^T = I
   \]
   But \( I^T = I \), since the identity matrix is symmetric:
   \[
   (A^{-1})^T A^T = I
   \]
   This shows that \( (A^{-1})^T \) is a left-inverse of \( A^T \).

3. **Similarly, transpose the other multiplication**:  
   \[
   (A^{-1} A)^T = I^T
   \]
   \[
   A^T (A^{-1})^T = I
   \]
   This shows that \( (A^{-1})^T \) is also a right-inverse of \( A^T \).

4. **Conclusion**:  
   Since \( (A^{-1})^T \) is both a left-inverse and a right-inverse of \( A^T \), it must be the inverse of \( A^T \). Thus, the inverse of \( A^T \), which is denoted by \( A^{-T} \), exists and is equal to \( (A^{-1})^T \).

Therefore, if \( A^{-1} \) exists, then \( A^{-T} \) exists.

### Part 2: **If \( A^{-T} \) exists, then \( A^{-1} \) exists.**

1. **Given**: \( A^{-T} \) exists.  
   - This means there is a matrix

Prove that A^{-1} (the inverse of matrix A) exists if and only if A^{-T} (the inverse of the transpose of A, also known as the transpose of A^{-1}) exists.

Explore the proof that the inverse of matrix A exists if and only if the inverse of the transpose of A exists. This problem covers key linear algebra concepts, including matrix inversion and properties of matrix transpose. Suitable for undergraduate students studying linear algebra.

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