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Question
If `A=[[-2],[4],[5]]` , B = [1 3 −6], verify that (AB)T = BT AT
Solution
\[Given: A = \begin{bmatrix}- 2 \\ 4 \\ 5\end{bmatrix}\]
\[ A^T = \begin{bmatrix}- 2 & 4 & 5\end{bmatrix}\]
\[B = \begin{bmatrix}1 & 3 & - 6\end{bmatrix} \]
\[ B^T = \begin{bmatrix}1 \\ 3 \\ - 6\end{bmatrix}\]
\[Now, \]
\[AB = \begin{bmatrix}- 2 \\ 4 \\ 5\end{bmatrix} \begin{bmatrix}1 & 3 & - 6\end{bmatrix} \]
\[ \Rightarrow AB = \begin{bmatrix}- 2 & - 6 & 12 \\ 4 & 12 & - 24 \\ 5 & 15 & - 30\end{bmatrix}\]
\[ \Rightarrow \left( AB \right)^T = \begin{bmatrix}- 2 & 4 & 5 \\ - 6 & 12 & 15 \\ 12 & - 24 & - 30\end{bmatrix} . . . \left( 1 \right)\]
\[ B^T A^T = \begin{bmatrix}1 \\ 3 \\ - 6\end{bmatrix}\begin{bmatrix}- 2 & 4 & 5\end{bmatrix}\]
\[ \Rightarrow B^T A^T = \begin{bmatrix}- 2 & 4 & 5 \\ - 6 & 12 & 15 \\ 12 & - 24 & - 30\end{bmatrix} . . . \left( 2 \right)\]
\[ \therefore \left( AB \right)^T = B^T A^T \left[ \text{From eqs} . (1) and (2) \right]\]
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