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Question
Let `A =[[2,-3],[-7,5]]` And `B=[[1,0],[2,-4]]` verify that
(2A)T = 2AT
Solution
\[Given: \hspace{0.167em} A = \begin{bmatrix}2 & - 3 \\ - 7 & 5\end{bmatrix}\]
\[ A^T = \begin{bmatrix}2 & - 7 \\ - 3 & 5\end{bmatrix}\]
\[B = \begin{bmatrix}1 & 0 \\ 2 & - 4\end{bmatrix} \]
\[ B^T = \begin{bmatrix}1 & 2 \\ 0 & - 4\end{bmatrix}\]
\[\left( i \right) \]
\[ \left( 2A \right)^T = 2 A^T \]
\[ \Rightarrow \left( 2\begin{bmatrix}2 & - 3 \\ - 7 & 5\end{bmatrix} \right)^T = 2\begin{bmatrix}2 & - 7 \\ - 3 & 5\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}4 & - 6 \\ - 14 & 10\end{bmatrix}^T = \begin{bmatrix}4 & - 14 \\ - 6 & 10\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}4 & - 14 \\ - 6 & 10\end{bmatrix} = \begin{bmatrix}4 & - 14 \\ - 6 & 10\end{bmatrix} \]
\[ \therefore LHS = RHS\]
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