Question

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

 (2A)^T = 2A^T

Answer 1

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

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

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

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

\[\left( i \right) \]

\[ \left( 2A \right)^T = 2 A^T \]

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

\[ \Rightarrow \begin{bmatrix}4 & - 6 \\ - 14 & 10\end{bmatrix}^T = \begin{bmatrix}4 & - 14 \\ - 6 & 10\end{bmatrix}\]

\[ \Rightarrow \begin{bmatrix}4 & - 14 \\ - 6 & 10\end{bmatrix} = \begin{bmatrix}4 & - 14 \\ - 6 & 10\end{bmatrix} \]

\[ \therefore LHS = RHS\]