To solve this problem, let’s verify each of the statements individually using properties of matrix transposition and inverses.

Part (a): $(A^T)^{-1} = (A^{-1})^T$

To prove this, let’s consider properties of transposes and inverses of matrices:

1. Property: If $A$ is an invertible (nonsingular) matrix, then $(A^{-1})^T$ is also invertible, and $(A^{-1})^T = (A^T)^{-1}$.

   This property holds because: $(A \cdot A^{-1})^T = I^T = I$ Expanding this, we get: $A^T \cdot (A^{-1})^T = I$ This equation implies that $(A^{-1})^T$ is the inverse of $A^T$, which means: $(A^T)^{-1} = (A^{-1})^T$

Therefore, Statement (a) is true.

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

To prove this, we use the properties of transposes and inverses for matrix products:

1. Property of Transposes: $(A \cdot B)^T = B^T \cdot A^T$.

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

2. Property of Inverses for Products: For any two invertible matrices $X$ and $Y$, we have $(X \cdot Y)^{-1} = Y^{-1} \cdot X^{-1}$.

   Applying this to $B^T \cdot A^T$, we get: $(B^T \cdot A^T)^{-1} = (A^T)^{-1} \cdot (B^T)^{-1}$

3. From part (a), we know that $(A^T)^{-1} = (A^{-1})^T$ and $(B^T)^{-1} = (B^{-1})^T$.

   Substituting these, we get: $(A^T)^{-1} \cdot (B^T)^{-1} = (A^{-1})^T \cdot (B^{-1})^T$

4. Using the property of transpose for products again, we have: $(A^{-1} \cdot B^{-1})^T = ((A \cdot B)^{-1})^T$

Therefore, Statement (b) is also true.

Summary

Both statements are true:

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

These follow from the properties of matrix transposition and inversion, specifically that the inverse of a transpose is the transpose of the inverse, and the inverse of a product is the product of the inverses in reverse order.

Would you like more details or have any questions?

Here are some related questions to explore further:

1. What other properties of transposes and inverses apply to block matrices?
2. How would these properties extend if $A$ and $B$ were not square matrices?
3. Can you show a concrete example using specific $2 \times 2$ matrices to verify these properties?
4. What is the relationship between the determinant of a matrix and its transpose?
5. How would these proofs change if $A$ and $B$ were complex matrices?

Tip: When working with matrix transposes and inverses, remember that operations on products are often non-commutative, so reversing the order of matrices in expressions like $(A \cdot B)^{-1}$ is essential.