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.