Verify that (AB)^T = B^TA^T

Given:

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

To Verify:

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

Step 1: Find AB

\[ AB = \begin{bmatrix} -2 \\ 4 \\ 5 \end{bmatrix} \begin{bmatrix} 1 & 3 & -6 \end{bmatrix} = \begin{bmatrix} -2 & -6 & 12 \\ 4 & 12 & -24 \\ 5 & 15 & -30 \end{bmatrix} \]

Step 2: Find (AB)^T

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

Step 3: Find A^T and B^T

\[ A^T = \begin{bmatrix} -2 & 4 & 5 \end{bmatrix}, \quad B^T = \begin{bmatrix} 1 \\ 3 \\ -6 \end{bmatrix} \]

Step 4: Find B^TA^T

\[ B^T A^T = \begin{bmatrix} 1 \\ 3 \\ -6 \end{bmatrix} \begin{bmatrix} -2 & 4 & 5 \end{bmatrix} = \begin{bmatrix} -2 & 4 & 5 \\ -6 & 12 & 15 \\ 12 & -24 & -30 \end{bmatrix} \]

Conclusion:

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

Hence Verified.