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Let $\mathrm{A}=\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ll}1 & \mathrm{x} \\ 0 & 1\end{array}\right]$
If $\mathrm{AB}=\mathrm{BA}$, then what is the value of $\mathrm{x}$?
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If $\mathrm{AB}=\mathrm{BA}$, then what is the value of $\mathrm{x}$?
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As given $\mathrm{A}=\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]$ and $\mathrm{B}=\left|\begin{array}{ll}1 & \mathrm{x} \\ 0 & 1\end{array}\right|$
$\mathrm{AB}=\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]\left[\begin{array}{ll}1 & \mathrm{x} \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}1 & \mathrm{x} \\ 0 & -1\end{array}\right]$
and $\mathrm{BA}=\left[\begin{array}{ll}1 & \mathrm{x} \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]=\left[\begin{array}{cc}1 & -\mathrm{x} \\ 0 & -1\end{array}\right]$
Given that $\mathrm{AB}=\mathrm{BA}$
We have, $\left[\begin{array}{cc}1 & \mathrm{x} \\ 0 & -1\end{array}\right]=\left[\begin{array}{cc}1 & -\mathrm{x} \\ 0 & -1\end{array}\right]$
$\Rightarrow \mathrm{x}=-\mathrm{x} \Rightarrow 2 \mathrm{x}=0 \Rightarrow \mathrm{x}=0$
$\mathrm{AB}=\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]\left[\begin{array}{ll}1 & \mathrm{x} \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}1 & \mathrm{x} \\ 0 & -1\end{array}\right]$
and $\mathrm{BA}=\left[\begin{array}{ll}1 & \mathrm{x} \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}1 & 0 \\ 0 & -1\end{array}\right]=\left[\begin{array}{cc}1 & -\mathrm{x} \\ 0 & -1\end{array}\right]$
Given that $\mathrm{AB}=\mathrm{BA}$
We have, $\left[\begin{array}{cc}1 & \mathrm{x} \\ 0 & -1\end{array}\right]=\left[\begin{array}{cc}1 & -\mathrm{x} \\ 0 & -1\end{array}\right]$
$\Rightarrow \mathrm{x}=-\mathrm{x} \Rightarrow 2 \mathrm{x}=0 \Rightarrow \mathrm{x}=0$
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