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Question:

If $A=\left[\begin{array}{rr}1 & -1 \\ -1 & 1\end{array}\right]$, satisfies the matrix equation $A^{2}=k A$, write the value of $k$.

Solution:

$A^{2}=A A$

$\Rightarrow A^{2}=\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right]\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right]$

$\Rightarrow A^{2}=\left[\begin{array}{cc}1+1 & -1-1 \\ -1-1 & 1+1\end{array}\right]$

$\Rightarrow A^{2}=\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]$

Now,

$A^{2}=k A$

$\Rightarrow\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]=k\left[\begin{array}{cc}1 & -1 \\ -1 & 1\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]=\left[\begin{array}{cc}k & -k \\ -k & k\end{array}\right]$

$\therefore k=2$

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