https://file.aiquickdraw.com/mathbot/file/1726587296416q1dnpqw0.jpg

The equation to prove from the image is:

\[
(A^{-1})^{-1} = A
\]

This statement involves matrix inverses. Here's how we can prove it:

### Step-by-Step Proof:

1. **Matrix Inverse Definition:**
   For any invertible matrix \( A \), its inverse \( A^{-1} \) satisfies the condition:
   \[
   A \cdot A^{-1} = I
   \]
   where \( I \) is the identity matrix.

2. **Taking the Inverse Again:**
   Now, consider \( (A^{-1})^{-1} \). This means taking the inverse of the inverse of \( A \). According to matrix theory, the inverse of the inverse matrix brings us back to the original matrix:
   \[
   (A^{-1})^{-1} = A
   \]

### Why This Works:
- If \( B = A^{-1} \), then \( B^{-1} = A \). Therefore, \( (A^{-1})^{-1} = A \).
  
Thus, we have proven that:

\[
(A^{-1})^{-1} = A
\]

Do you need more details on matrix inverses or the proof? Here are five related questions for further practice:

1. What is the inverse of a 2x2 matrix, and how is it calculated?
2. Can every matrix be inverted? Why or why not?
3. What is the relationship between the determinant of a matrix and its invertibility?
4. How does the identity matrix behave in matrix multiplication?
5. How do we compute the inverse of a matrix using row reduction?

**Tip:** When proving matrix properties, always recall the identity matrix \( I \), as it plays a central role in matrix inversions.

This solution proves the matrix inverse identity (A^{-1})^{-1} = A using key concepts in matrix theory and inverse matrices. The step-by-step explanation involves the property of inverse matrices and the identity matrix. Suitable for undergraduate-level students studying linear algebra.

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