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This equation presents the property of the transpose of a matrix product, stating that the transpose of the product of two matrices \( A \) and \( B \) is equal to the product of their individual transposes in reverse order:

\[
(A \cdot B)^T = B^T \cdot A^T
\]

### Proof:

Let \( A \) and \( B \) be two matrices such that their product \( A \cdot B \) is defined (i.e., the number of columns of \( A \) equals the number of rows of \( B \)).

1. **Transpose definition**: 
   The transpose of a matrix is defined by switching its rows and columns. Formally, for any matrix \( M \), the element at position \( (i, j) \) of \( M^T \) is the element at position \( (j, i) \) of \( M \). In symbols:
   \[
   (M^T)_{ij} = M_{ji}
   \]

2. **Element of product**:
   The element at position \( (i, j) \) in the matrix product \( A \cdot B \) is given by:
   \[
   (A \cdot B)_{ij} = \sum_k A_{ik} B_{kj}
   \]
   This is the sum of the products of corresponding elements from the \( i \)-th row of \( A \) and the \( j \)-th column of \( B \).

3. **Transpose of the product**:
   To find \( (A \cdot B)^T \), we transpose the matrix \( A \cdot B \), which means the element at position \( (i, j) \) in \( A \cdot B \) becomes the element at position \( (j, i) \) in \( (A \cdot B)^T \):
   \[
   (A \cdot B)^T_{ij} = (A \cdot B)_{ji}
   \]
   Using the expression for \( (A \cdot B)_{ji} \), we get:
   \[
   (A \cdot B)^T_{ij} = \sum_k A_{jk} B_{ki}
   \]

4. **Transpose of individual matrices**:
   Now let's consider the product \( B^T \cdot A^T \). The element at position \( (i, j) \) in \( B^T \cdot A^T \) is given by:
   \[
   (B^T \cdot A^T)_{ij} = \sum_k B^T_{ik} A^T_{kj}
   \]
   But by the definition of the transpose:
   \[
   B^T_{ik} = B_{ki} \quad \text{and} \quad A^T_{kj} = A_{jk}
   \]
   Therefore:
   \[
   (B^T \cdot A^T)_{ij} = \sum_k B_{ki} A_{jk}
   \]
   which is exactly the same expression as \( (A \cdot B)^T_{ij} \).

Since both expressions for \( (A \cdot B)^T \) and \( B^T \cdot A^T \) are identical, we have:
\[
(A \cdot B)^T = B^T \cdot A^T
\]

Thus, the property is proven.

Would you like further details or have any questions? Here are some related questions:

1. Can the transpose property be extended to more than two matrices?
2. What is the physical significance of this property in applications like linear transformations?
3. How does the proof change if matrices \( A \) and \( B \) are not square?
4. Is there a similar property for the inverse of a matrix product?
5. How do you compute the transpose of a matrix product in software like MATLAB or Python?

**Tip**: When dealing with matrices, always ensure dimensions are compatible before performing operations like multiplication or transpose.

This page explains how to prove the matrix product transpose property, which states that the transpose of the product of two matrices A and B is equal to the product of their individual transposes in reverse order. Includes a step-by-step breakdown of the proof, suitable for undergraduate students.

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