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**Problem Statement:**
Verify that \((AB)^T = B^T A^T\) for the following matrices:

**(i)** 
\[ A = \begin{bmatrix} 1 \\ 3 \\ 6 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 4 & 5 \end{bmatrix} \]

### Solution for (i):

1. **Compute \( AB \):**
   Since \( A \) is a \(3 \times 1\) matrix and \( B \) is a \(1 \times 3\) matrix, their product \( AB \) will be a \(3 \times 3\) matrix where each element of \( A \) is multiplied by each element of \( B \):
   \[ AB = \begin{bmatrix} 1 \\ 3 \\ 6 \end{bmatrix} \begin{bmatrix} 2 & 4 & 5 \end{bmatrix} = \begin{bmatrix} 1 \cdot 2 & 1 \cdot 4 & 1 \cdot 5 \\ 3 \cdot 2 & 3 \cdot 4 & 3 \cdot 5 \\ 6 \cdot 2 & 6 \cdot 4 & 6 \cdot 5 \end{bmatrix} = \begin{bmatrix} 2 & 4 & 5 \\ 6 & 12 & 15 \\ 12 & 24 & 30 \end{bmatrix} \]

2. **Transpose \( AB \):**
   \[ (AB)^T = \begin{bmatrix} 2 & 6 & 12 \\ 4 & 12 & 24 \\ 5 & 15 & 30 \end{bmatrix} \]

3. **Compute \( B^T \) and \( A^T \):**
   \[ B^T = \begin{bmatrix} 2 \\ 4 \\ 5 \end{bmatrix}, \quad A^T = \begin{bmatrix} 1 & 3 & 6 \end{bmatrix} \]

4. **Compute \( B^T A^T \):**
   \[ B^T A^T = \begin{bmatrix} 2 \\ 4 \\ 5 \end{bmatrix} \begin{bmatrix} 1 & 3 & 6 \end{bmatrix} = \begin{bmatrix} 2 \cdot 1 & 2 \cdot 3 & 2 \cdot 6 \\ 4 \cdot 1 & 4 \cdot 3 & 4 \cdot 6 \\ 5 \cdot 1 & 5 \cdot 3 & 5 \cdot 6 \end{bmatrix} = \begin{bmatrix} 2 & 6 & 12 \\ 4 & 12 & 24 \\ 5 & 15 & 30 \end{bmatrix} \]

### Conclusion:
\((AB)^T\) and \(B^T A^T\) are identical, verifying the identity \((AB)^T = B^T A^T\) for this set of matrices.

Would you like further details or help with another matrix verification? Let me know if you need assistance with any other parts from the uploaded image!

1. Would you like to see a step-by-step solution for part (ii) of the problem?
2. Do you need the calculation for a specific matrix operation like multiplication or transposition?
3. Are you interested in understanding why the property \((AB)^T = B^T A^T\) holds true for matrices?
4. Do you want to explore how matrix dimensions affect the ability to multiply two matrices?
5. Would you like a refresher on matrix notation and terminology?

**Tip:** When working with matrix multiplication and transpositions, always check the dimension compatibility and keep track of the order of operations, as it significantly affects the outcome.

Verify (AB)^T = B^T A^T for the following matrices.

Verify the matrix transpose identity (AB)^T = B^T A^T using the given matrices. The problem involves matrix multiplication and transposition, covering key concepts from linear algebra. Suitable for Grades 10-12.

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