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If $\mathrm{A}=\left[\begin{array}{ll}\alpha & 0 \\ 1 & 1\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right]$
such that $A^{2}=B$, then what is the value of $\alpha$?
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such that $A^{2}=B$, then what is the value of $\alpha$?
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1
Let $\mathrm{A}=\left[\begin{array}{ll}\alpha & 0 \\ 1 & 1\end{array}\right]$
$\Rightarrow \quad \mathrm{A}^{2}=\left[\begin{array}{ll}\alpha & 0 \\ 1 & 1\end{array}\right]\left[\begin{array}{ll}\alpha & 0 \\ 1 & 1\end{array}\right]$
$\Rightarrow \quad \mathrm{A}^{2}=\left[\begin{array}{cc}\alpha^{2} & 0 \\ \alpha+1 & 1\end{array}\right]$
But it is given that
$A^{2}=B$
$\Rightarrow \quad\left[\begin{array}{ll}\alpha^{2} & 0 \\ \alpha+1 & 1\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right]$
$\Rightarrow \quad \alpha+1=2$
$\Rightarrow \quad \alpha=1$
$\Rightarrow \quad \mathrm{A}^{2}=\left[\begin{array}{ll}\alpha & 0 \\ 1 & 1\end{array}\right]\left[\begin{array}{ll}\alpha & 0 \\ 1 & 1\end{array}\right]$
$\Rightarrow \quad \mathrm{A}^{2}=\left[\begin{array}{cc}\alpha^{2} & 0 \\ \alpha+1 & 1\end{array}\right]$
But it is given that
$A^{2}=B$
$\Rightarrow \quad\left[\begin{array}{ll}\alpha^{2} & 0 \\ \alpha+1 & 1\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 2 & 1\end{array}\right]$
$\Rightarrow \quad \alpha+1=2$
$\Rightarrow \quad \alpha=1$
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