Verify that (AB)T = BTAT
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 AT and BT
\[ A^T = \begin{bmatrix} -2 & 4 & 5 \end{bmatrix}, \quad B^T = \begin{bmatrix} 1 \\ 3 \\ -6 \end{bmatrix} \]
Step 4: Find BTAT
\[ B^T A^T = \begin{bmatrix} 1 \\ 3 \\ -6 \end{bmatrix} \begin{bmatrix} -2 & 4 & 5 \end{bmatrix} = \begin{bmatrix} -2 & 4 & 5 \\ -6 & 12 & 15 \\ 12 & -24 & -30 \end{bmatrix} \]
Conclusion:
\[ (AB)^T = B^T A^T \]
Hence Verified.