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If matrix $A=\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]$ and $A^{2}=p A$, then the value of $p$ is
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Verified Answer
The correct answer is:
4
Given, $A=\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right] \Rightarrow A^{2}=\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]$
$$
=\left[\begin{array}{cc}
4+4 & -4-4 \\
-4-4 & 4+4
\end{array}\right]=\left[\begin{array}{cc}
8 & -8 \\
-8 & 8
\end{array}\right]
$$
Given that, $A^{2}=p A$
$$
\Rightarrow\left[\begin{array}{cc}
8 & -8 \
-8 & 8
\end{array}\right]=p\left[\begin{array}{cc}
2 & -2 \\
-2 & 2
\end{array}\right] \Rightarrow\left[\begin{array}{cc}
8 & -8 \\
-8 & 8
\end{array}\right]=\left[\begin{array}{cc}
2 p & -2 p \\
-2 p & 2 p
\end{array}\right]
$$
Now, $\quad 2 p=8 \Rightarrow p=4$
$$
=\left[\begin{array}{cc}
4+4 & -4-4 \\
-4-4 & 4+4
\end{array}\right]=\left[\begin{array}{cc}
8 & -8 \\
-8 & 8
\end{array}\right]
$$
Given that, $A^{2}=p A$
$$
\Rightarrow\left[\begin{array}{cc}
8 & -8 \
-8 & 8
\end{array}\right]=p\left[\begin{array}{cc}
2 & -2 \\
-2 & 2
\end{array}\right] \Rightarrow\left[\begin{array}{cc}
8 & -8 \\
-8 & 8
\end{array}\right]=\left[\begin{array}{cc}
2 p & -2 p \\
-2 p & 2 p
\end{array}\right]
$$
Now, $\quad 2 p=8 \Rightarrow p=4$
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