Answer
$A$ and $B$ are inverses of each other.
Work Step by Step
The inverse of an $n\times n$ matrix $A$ is that $n\times n$ matrix $A^{-1}$ which,
when multiplied by $A$ on either side, yields the $n\times n$ identity matrix $I$.
$A A^{-1}=A^{-1}A=I.$
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$AB=\left[\begin{array}{ll}
0 & 1\\
1 & 0
\end{array}\right]\left[\begin{array}{ll}
0 & 1\\
1 & 0
\end{array}\right]=\left[\begin{array}{ll}
0+1 & 0+0\\
0+0 & 1+0
\end{array}\right]=\left[\begin{array}{ll}
1 & 0\\
0 & 1
\end{array}\right]=I_{2}$
$BA=\left[\begin{array}{ll}
0 & 1\\
1 & 0
\end{array}\right]\left[\begin{array}{ll}
0 & 1\\
1 & 0
\end{array}\right]=\left[\begin{array}{ll}
0+1 & 0+0\\
0+0 & 1+0
\end{array}\right]=\left[\begin{array}{ll}
1 & 0\\
0 & 1
\end{array}\right]=I_{2}$
$A$ and $B$ are inverses of each other.