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The problem requires verification of the matrix identity \((AB)^T = B^T A^T\), where matrices \( A \) and \( B \) are given as:

\[
A = \begin{bmatrix} 2 & 5 & 6 \\ 0 & 1 & 2 \end{bmatrix}
\quad \text{and} \quad
B = \begin{bmatrix} 6 & 0 & 4 \\ 5 & 7 & 0 \end{bmatrix}
\]

### Step 1: Compute \( AB \)
First, we calculate the product \( AB \):

\[
AB = \begin{bmatrix} 2 & 5 & 6 \\ 0 & 1 & 2 \end{bmatrix}
    \times
    \begin{bmatrix} 6 & 0 & 4 \\ 5 & 7 & 0 \end{bmatrix}
\]

This multiplication can be done by taking the dot product of rows from \( A \) with columns from \( B \):

\[
AB = \begin{bmatrix} 
(2 \times 6 + 5 \times 5) & (2 \times 0 + 5 \times 7) & (2 \times 4 + 5 \times 0) \\
(0 \times 6 + 1 \times 5) & (0 \times 0 + 1 \times 7) & (0 \times 4 + 1 \times 0)
\end{bmatrix}
\]

\[
AB = \begin{bmatrix} 
47 & 35 & 8 \\
5 & 7 & 0 
\end{bmatrix}
\]

### Step 2: Compute \( (AB)^T \)
Now, compute the transpose of the product \( AB \):

\[
(AB)^T = \begin{bmatrix} 
47 & 35 & 8 \\
5 & 7 & 0 
\end{bmatrix}^T
=
\begin{bmatrix} 
47 & 5 \\
35 & 7 \\
8 & 0
\end{bmatrix}
\]

### Step 3: Compute \( B^T \) and \( A^T \)
Next, we calculate the transposes of \( B \) and \( A \):

\[
B^T = \begin{bmatrix} 
6 & 5 \\
0 & 7 \\
4 & 0
\end{bmatrix},
\quad
A^T = \begin{bmatrix} 
2 & 0 \\
5 & 1 \\
6 & 2
\end{bmatrix}
\]

### Step 4: Compute \( B^T A^T \)
Now, calculate the product \( B^T A^T \):

\[
B^T A^T = \begin{bmatrix} 6 & 5 \\ 0 & 7 \\ 4 & 0 \end{bmatrix} 
\times 
\begin{bmatrix} 2 & 0 \\ 5 & 1 \\ 6 & 2 \end{bmatrix}
\]

\[
B^T A^T = \begin{bmatrix} 
(6 \times 2 + 5 \times 5) & (6 \times 0 + 5 \times 1) \\
(0 \times 2 + 7 \times 5) & (0 \times 0 + 7 \times 1) \\
(4 \times 2 + 0 \times 5) & (4 \times 0 + 0 \times 1)
\end{bmatrix}
\]

\[
B^T A^T = \begin{bmatrix} 
47 & 5 \\
35 & 7 \\
8 & 0
\end{bmatrix}
\]

### Step 5: Conclusion
Since \( (AB)^T = B^T A^T \), the given matrix identity is verified.

Would you like more details or have any further questions?

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Here are five related questions for further exploration:
1. What are the conditions under which \((AB)^T = B^T A^T\) always holds?
2. How does the transpose of a matrix product differ from the product of transposes?
3. Can this property of transposes be extended to three or more matrices?
4. How does the inverse of a product of matrices relate to their individual inverses?
5. How does matrix multiplication relate to transformations in geometry?

**Tip:** When multiplying matrices, make sure the dimensions are compatible—i.e., the number of columns in the first matrix must match the number of rows in the second matrix.

Verify that (AB)^T = B^T A^T, where A = [[2, 5, 6], [0, 1, 2]] and B = [[6, 0, 4], [5, 7, 0]].

This problem verifies the matrix identity (AB)^T = B^T A^T using matrix multiplication and transposition. A step-by-step guide through the calculation process demonstrates how to multiply matrices and apply the transpose operation. Suitable for students in Grades 10-12.

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