Question

Let  `A =[[2,-3],[-7,5]]` And `B=[[1,0],[2,-4]]` verify that 

(A − B)^T = A^T − B^T

Answer 1

\[Given: \hspace{0.167em} A = \begin{bmatrix}2 & - 3 \\ - 7 & 5\end{bmatrix}\]

\[ A^T = \begin{bmatrix}2 & - 7 \\ - 3 & 5\end{bmatrix}\]

\[\]

\[B = \begin{bmatrix}1 & 0 \\ 2 & - 4\end{bmatrix} \]

\[ B^T = \begin{bmatrix}1 & 2 \\ 0 & - 4\end{bmatrix}\]

\[\left( iii \right) \left( A - B \right)^T = A^T - B^T \]

\[ \Rightarrow \left( \begin{bmatrix}2 & - 3 \\ - 7 & 5\end{bmatrix} - \begin{bmatrix}1 & 0 \\ 2 & - 4\end{bmatrix} \right)^T = \begin{bmatrix}2 & - 7 \\ - 3 & 5\end{bmatrix} - \begin{bmatrix}1 & 2 \\ 0 & - 4\end{bmatrix}\]

\[ \Rightarrow \left( \begin{bmatrix}2 - 1 & - 3 - 0 \\ - 7 - 2 & 5 + 4\end{bmatrix} \right)^T = \begin{bmatrix}2 - 1 & - 7 - 2 \\ - 3 - 0 & 5 + 4\end{bmatrix}\]

\[ \Rightarrow \left( \begin{bmatrix}1 & - 3 \\ - 9 & 9\end{bmatrix} \right)^T = \begin{bmatrix}1 & - 9 \\ - 3 & 9\end{bmatrix}\]

\[ \Rightarrow \begin{bmatrix}1 & - 9 \\ - 3 & 9\end{bmatrix} = \begin{bmatrix}1 & - 9 \\ - 3 & 9\end{bmatrix}\]

\[ \therefore LHS = RHS\]