Question
Let $\text{A}=\begin{bmatrix}2&-3\\-7&5\end{bmatrix}$ and $\text{B}=\begin{bmatrix}1&0\\2&-4\end{bmatrix},$ verify that
$(2\text{A})^\text{T}=2\text{A}^\text{T}$
$(2\text{A})^\text{T}=2\text{A}^\text{T}$
✓
Answer
Given: $\text{A}=\begin{bmatrix}2&-3\\-7&5\end{bmatrix}$
$\text{A}^\text{T}=\begin{bmatrix}2&-7\\-3&5\end{bmatrix}$
$\text{B}=\begin{bmatrix}1&0\\2&-4\end{bmatrix}$
$\text{A}^\text{T}=\begin{bmatrix}1&2\\0&-4\end{bmatrix}$
$(2\text{A})^\text{T}=2\text{A}^\text{T}$
$\Rightarrow\begin{pmatrix} 2\begin{bmatrix}2&-3\\-7&5\end{bmatrix}\end{pmatrix}^\text{T}=2\begin{bmatrix} 2&-7\\-3&5\end{bmatrix}$
$\Rightarrow\begin{bmatrix} 4&-6\\-14&10\end{bmatrix}^\text{T}=\begin{bmatrix}4&-14\\-6&10 \end{bmatrix}$
$\Rightarrow\begin{bmatrix}4&-14\\-6&10\end{bmatrix}=\begin{bmatrix}4&-14\\-6&10\end{bmatrix}$
$\therefore\ \text{LHS}=\text{RHS}$
$\text{A}^\text{T}=\begin{bmatrix}2&-7\\-3&5\end{bmatrix}$
$\text{B}=\begin{bmatrix}1&0\\2&-4\end{bmatrix}$
$\text{A}^\text{T}=\begin{bmatrix}1&2\\0&-4\end{bmatrix}$
$(2\text{A})^\text{T}=2\text{A}^\text{T}$
$\Rightarrow\begin{pmatrix} 2\begin{bmatrix}2&-3\\-7&5\end{bmatrix}\end{pmatrix}^\text{T}=2\begin{bmatrix} 2&-7\\-3&5\end{bmatrix}$
$\Rightarrow\begin{bmatrix} 4&-6\\-14&10\end{bmatrix}^\text{T}=\begin{bmatrix}4&-14\\-6&10 \end{bmatrix}$
$\Rightarrow\begin{bmatrix}4&-14\\-6&10\end{bmatrix}=\begin{bmatrix}4&-14\\-6&10\end{bmatrix}$
$\therefore\ \text{LHS}=\text{RHS}$
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