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Question
If `A= [[3],[5],[2]]` And B=[1 0 4] , Verify that `(AB)^T=B^TA^T`
Solution
\[Given: \hspace{0.167em} A = \begin{bmatrix}3 \\ 5 \\ 2\end{bmatrix}\]
\[ A^T = \begin{bmatrix}3 & 5 & 2\end{bmatrix}\]
\[\]\[B = \begin{bmatrix}1 & 0 & 4\end{bmatrix}\]
\[\]\[ B^T = \begin{bmatrix}1 \\ 0 \\ 4\end{bmatrix}\]
\[\]\[Now, \]
\[AB = \begin{bmatrix}3 \\ 5 \\ 2\end{bmatrix}\begin{bmatrix}1 & 0 & 4\end{bmatrix}\]
\[ \Rightarrow AB = \begin{bmatrix}3 & 0 & 12 \\ 5 & 0 & 20 \\ 2 & 0 & 8\end{bmatrix}\]
\[ \Rightarrow \left( AB \right)^T = \begin{bmatrix}3 & 5 & 2 \\ 0 & 0 & 0 \\ 12 & 20 & 8\end{bmatrix} . . . \left( 1 \right)\]
\[\]
\[ B^T A^T = \begin{bmatrix}1 \\ 0 \\ 4\end{bmatrix}\begin{bmatrix}3 & 5 & 2\end{bmatrix}\]
\[ \Rightarrow B^T A^T = \begin{bmatrix}3 & 5 & 2 \\ 0 & 0 & 0 \\ 12 & 20 & 8\end{bmatrix} . . . \left( 2 \right)\]
\[\]
\[ \Rightarrow \left( AB \right)^T = B^T A^T \left[ \text{From eqs }. (1) \hspace{0.167em} \text{and (2)} \right]\]
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