Given that $A$ and $B$ are invertible matrices, we want to express $\left( (AB)^{-1} \right)^T$ in terms of $A^{-1}$ and $B^{-1}$.

Steps:

1. Matrix Inversion Property:

   • The inverse of a product of matrices follows the rule: $(AB)^{-1} = B^{-1}A^{-1}$ This means that the inverse of a product is the reverse product of the inverses.

2. Transpose of a Matrix:

   • The transpose of a product of matrices follows the rule: $(AB)^T = B^T A^T$ However, we need the transpose of the inverse, not just the product.

3. Transpose of an Inverse:

   • The transpose of the inverse of a matrix is the inverse of the transpose: $\left( (AB)^{-1} \right)^T = \left( B^{-1}A^{-1} \right)^T$ By the property of the transpose of a product, we reverse the order of the matrices: $\left( B^{-1}A^{-1} \right)^T = (A^{-1})^T (B^{-1})^T$

Final Expression:

$\left( (AB)^{-1} \right)^T = (A^{-1})^T (B^{-1})^T$

Would you like more details on the matrix properties used here or have any questions?

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Here are some related questions:

1. What is the inverse of a transpose for a general matrix?
2. How do the determinant and transpose properties relate for invertible matrices?
3. How do we calculate the inverse of a matrix step-by-step?
4. What is the property of transposes when applied to sums of matrices?
5. Can a non-square matrix have an inverse or a transpose?

Tip: The inverse of a product reverses the order of the factors; be mindful of this when dealing with multiple matrices.