To determine whether the given statement

$((A \cdot B)^T)^{-1} = ((A \cdot B)^{-1})^T$

is true, let's go step by step:

Step 1: Recall Key Matrix Properties

1. Transposition of a Product: For any matrices $A$ and $B$ (of compatible dimensions), we have: $(A \cdot B)^T = B^T \cdot A^T$

2. Inverse of a Product: For invertible matrices $A$ and $B$, we have: $(A \cdot B)^{-1} = B^{-1} \cdot A^{-1}$

3. Inverse of a Transpose: For any invertible matrix $M$, we have: $(M^T)^{-1} = (M^{-1})^T$

Step 2: Analyze the Left-Hand Side (LHS)

The LHS is: $((A \cdot B)^T)^{-1}$

Using the property of the transpose of a product: $(A \cdot B)^T = B^T \cdot A^T$

Substitute this into the LHS: $((A \cdot B)^T)^{-1} = (B^T \cdot A^T)^{-1}$

Now apply the property of the inverse of a product: $(B^T \cdot A^T)^{-1} = (A^T)^{-1} \cdot (B^T)^{-1}$

Finally, use the property of the inverse of a transpose: $(A^T)^{-1} = (A^{-1})^T \quad \text{and} \quad (B^T)^{-1} = (B^{-1})^T$

Thus: $((A \cdot B)^T)^{-1} = (A^{-1})^T \cdot (B^{-1})^T$

Step 3: Analyze the Right-Hand Side (RHS)

The RHS is: $((A \cdot B)^{-1})^T$

Using the property of the inverse of a product: $(A \cdot B)^{-1} = B^{-1} \cdot A^{-1}$

Substitute this into the RHS: $((A \cdot B)^{-1})^T = (B^{-1} \cdot A^{-1})^T$

Now apply the property of the transpose of a product: $(B^{-1} \cdot A^{-1})^T = (A^{-1})^T \cdot (B^{-1})^T$

Step 4: Compare the Two Sides

From Step 2, the LHS is: $((A \cdot B)^T)^{-1} = (A^{-1})^T \cdot (B^{-1})^T$

From Step 3, the RHS is: $((A \cdot B)^{-1})^T = (A^{-1})^T \cdot (B^{-1})^T$

Thus, the LHS and RHS are equal.

Conclusion

The given statement is true: $((A \cdot B)^T)^{-1} = ((A \cdot B)^{-1})^T$

Would you like a more detailed explanation of any step? Here are some questions to explore further:

1. What are other properties of matrix transposes and inverses?
2. Can this result be generalized to more than two matrices?
3. How does this proof work when $A$ or $B$ is singular?
4. Why is the order of multiplication reversed in transposes and inverses?
5. Can you verify this with numerical examples?

Tip: Understanding matrix properties like these is essential in fields such as linear algebra, computer graphics, and machine learning!