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Question:
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Ifthe matrix
$A=\left[\begin{array}{ll}
\alpha & \beta \\
\beta & \alpha
\end{array}\right]$
is such that $A^{2}=I$, then which one of the following is correct?
Options:
$A=\left[\begin{array}{ll}
\alpha & \beta \\
\beta & \alpha
\end{array}\right]$
is such that $A^{2}=I$, then which one of the following is correct?
Solution:
1741 Upvotes
Verified Answer
The correct answer is:
$\alpha=0, \beta=1$ or $\alpha=1, \beta=0$
$\begin{array}{ll}\text { Let } & A=\left[\begin{array}{ll}\alpha & \beta \\ \beta & \alpha\end{array}\right] \\ \therefore & \mathrm{A}^{2}=\left[\begin{array}{ll}\alpha & \beta \\ \beta & \alpha\end{array}\right]\left[\begin{array}{ll}\alpha & \beta \\ \beta & \alpha\end{array}\right] \\ & =\left[\begin{array}{rc}\alpha^{2}+\beta^{2} & 2 \alpha \beta \\ 2 \alpha \beta & \alpha^{2}+\beta^{2}\end{array}\right]\end{array}$
Now, $\mathrm{A}^{2}=\mathrm{I}$
$\Rightarrow \quad\left[\begin{array}{cc}\alpha^{2}+\beta^{2} & 2 \alpha \beta \\ 2 \alpha \beta & \alpha^{2}+\beta^{2}\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$\Rightarrow \quad \alpha^{2}+\beta^{2}=1, \quad \alpha \beta=0$
$\Rightarrow \quad \alpha=0, \beta=1$
or $\quad \beta=0, \alpha=1$
Now, $\mathrm{A}^{2}=\mathrm{I}$
$\Rightarrow \quad\left[\begin{array}{cc}\alpha^{2}+\beta^{2} & 2 \alpha \beta \\ 2 \alpha \beta & \alpha^{2}+\beta^{2}\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$\Rightarrow \quad \alpha^{2}+\beta^{2}=1, \quad \alpha \beta=0$
$\Rightarrow \quad \alpha=0, \beta=1$
or $\quad \beta=0, \alpha=1$
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