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Question
Solution
\[A^2 = AA\]
\[ \Rightarrow A^2 = \begin{bmatrix}1 & - 1 \\ - 1 & 1\end{bmatrix}\begin{bmatrix}1 & - 1 \\ - 1 & 1\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}1 + 1 & - 1 - 1 \\ - 1 - 1 & 1 + 1\end{bmatrix}\]
\[ \Rightarrow A^2 = \begin{bmatrix}2 & - 2 \\ - 2 & 2\end{bmatrix}\]
\[Now, \]
\[ A^2 = kA\]
\[ \Rightarrow \begin{bmatrix}2 & - 2 \\ - 2 & 2\end{bmatrix} = k\begin{bmatrix}1 & - 1 \\ - 1 & 1\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}2 & - 2 \\ - 2 & 2\end{bmatrix} = \begin{bmatrix}k & - k \\ - k & k\end{bmatrix}\]
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