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Question
Let `A =[[2,-3],[-7,5]]` And `B=[[1,0],[2,-4]]` verify that
(AB)T = BT AT
Solution
\[Given: \hspace{0.167em} A = \begin{bmatrix}2 & - 3 \\ - 7 & 5\end{bmatrix}\]
\[ A^T = \begin{bmatrix}2 & - 7 \\ - 3 & 5\end{bmatrix}\]
\[\]
\[B = \begin{bmatrix}1 & 0 \\ 2 & - 4\end{bmatrix} \]
\[\left( iv \right) \left( AB \right)^T = B^T A^T \]
\[ \Rightarrow \left( \begin{bmatrix}2 & - 3 \\ - 7 & 5\end{bmatrix}\begin{bmatrix}1 & 0 \\ 2 & - 4\end{bmatrix} \right)^T = \begin{bmatrix}1 & 2 \\ 0 & - 4\end{bmatrix} \begin{bmatrix}2 & - 7 \\ - 3 & 5\end{bmatrix}\]
\[ \Rightarrow \left( \begin{bmatrix}2 - 6 & 0 + 12 \\ - 7 + 10 & 0 - 20\end{bmatrix} \right)^T = \begin{bmatrix}2 - 6 & - 7 + 10 \\ 0 + 12 & 0 - 20\end{bmatrix}\]
\[ \Rightarrow \left( \begin{bmatrix}- 4 & 12 \\ 3 & - 20\end{bmatrix} \right)^T = \begin{bmatrix}- 4 & 3 \\ 12 & - 20\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}- 4 & 3 \\ 12 & - 20\end{bmatrix} = \begin{bmatrix}- 4 & 3 \\ 12 & - 20\end{bmatrix}\]
\[ \therefore LHS = RHS\]
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