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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
\[Given: \hspace{0.167em} A = \begin{bmatrix}2 & - 3 \\ - 7 & 5\end{bmatrix}\]
\[ A^T = \begin{bmatrix}2 & - 7 \\ - 3 & 5\end{bmatrix}\]
\[\]
\[B = \begin{bmatrix}1 & 0 \\ 2 & - 4\end{bmatrix} \]
\[ B^T = \begin{bmatrix}1 & 2 \\ 0 & - 4\end{bmatrix}\]
\[\left( ii \right) \left( A + B \right)^T = A^T + B^T \]
\[ \Rightarrow \left( \begin{bmatrix}2 & - 3 \\ - 7 & 5\end{bmatrix} + \begin{bmatrix}1 & 0 \\ 2 & - 4\end{bmatrix} \right)^T = \begin{bmatrix}2 & - 7 \\ - 3 & 5\end{bmatrix} + \begin{bmatrix}1 & 2 \\ 0 & - 4\end{bmatrix}\]
\[ \Rightarrow \left( \begin{bmatrix}2 + 1 & - 3 + 0 \\ - 7 + 2 & 5 - 4\end{bmatrix} \right)^T = \begin{bmatrix}2 + 1 & - 7 + 2 \\ - 3 + 0 & 5 - 4\end{bmatrix}\]
\[ \Rightarrow \left( \begin{bmatrix}3 & - 3 \\ - 5 & 1\end{bmatrix} \right)^T = \begin{bmatrix}3 & - 5 \\ - 3 & 1\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}3 & - 5 \\ - 3 & 1\end{bmatrix} = \begin{bmatrix}3 & - 5 \\ - 3 & 1\end{bmatrix}\]
\[ \therefore LHS = RHS\]
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