If A and B are invertible, express [(AB)
−1
]
T with A−1 and B−1
.

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.

If A and B are invertible, express [(AB)^{-1}]^T with A^{-1} and B^{-1}.

Learn how to express the transpose of the inverse of a product of matrices [(AB)^{-1}]^T using the inverses of individual matrices A^{-1} and B^{-1}. This detailed step-by-step explanation covers matrix inversion, transpose rules, and essential linear algebra concepts. Suitable for college-level students.

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