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Question
Let `A =[[2,-3],[-7,5]]` And `B=[[1,0],[2,-4]]` verify that
(A − B)T = AT − BT
Solution
\[Given: \hspace{0.167em} A = \begin{bmatrix}2 & - 3 \\ - 7 & 5\end{bmatrix}\]
\[ A^T = \begin{bmatrix}2 & - 7 \\ - 3 & 5\end{bmatrix}\]
\[\]
\[B = \begin{bmatrix}1 & 0 \\ 2 & - 4\end{bmatrix} \]
\[ B^T = \begin{bmatrix}1 & 2 \\ 0 & - 4\end{bmatrix}\]
\[\left( iii \right) \left( A - B \right)^T = A^T - B^T \]
\[ \Rightarrow \left( \begin{bmatrix}2 & - 3 \\ - 7 & 5\end{bmatrix} - \begin{bmatrix}1 & 0 \\ 2 & - 4\end{bmatrix} \right)^T = \begin{bmatrix}2 & - 7 \\ - 3 & 5\end{bmatrix} - \begin{bmatrix}1 & 2 \\ 0 & - 4\end{bmatrix}\]
\[ \Rightarrow \left( \begin{bmatrix}2 - 1 & - 3 - 0 \\ - 7 - 2 & 5 + 4\end{bmatrix} \right)^T = \begin{bmatrix}2 - 1 & - 7 - 2 \\ - 3 - 0 & 5 + 4\end{bmatrix}\]
\[ \Rightarrow \left( \begin{bmatrix}1 & - 3 \\ - 9 & 9\end{bmatrix} \right)^T = \begin{bmatrix}1 & - 9 \\ - 3 & 9\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}1 & - 9 \\ - 3 & 9\end{bmatrix} = \begin{bmatrix}1 & - 9 \\ - 3 & 9\end{bmatrix}\]
\[ \therefore LHS = RHS\]
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