  We present an explicit formula of the powers for the $2\times 2$ quantum
matrices, that is a natural quantum analogue of the powers of the usual
$2\times 2$ matrices. As applications, we give some non-commutative relations
of the entries of the powers for the $2\times 2$ quantum matrices, which is a
simple proof of the results of Vokos-Zumino-Wess (1990).
