This statement is false.

While matrix $A$ has an inverse, the inverse of the transpose of $A$, denoted $(A^T)^{-1}$, is not necessarily the same as $A^{-1}$. In fact, there is a specific relationship between the inverse of a matrix and the inverse of its transpose, and it is:

$(A^T)^{-1} = (A^{-1})^T$

So, the inverse of the transpose of $A$ is the transpose of the inverse of $A$, but it is not necessarily the same matrix as $A^{-1}$.

Would you like more details or further clarification?

Here are 5 related questions to expand on this concept:

1. What are the conditions for a matrix to have an inverse?
2. How can we prove that $(A^T)^{-1} = (A^{-1})^T$?
3. What properties do orthogonal matrices have regarding transposition and inversion?
4. Can the product of two invertible matrices be inverted? If so, how?
5. What is the inverse of a diagonal matrix?

Tip: When dealing with matrix transpositions and inverses, always remember that the operations on the matrix elements often "switch" their order, especially in products.