Matrix presentations of braids and applications

We show that there exists a one-to-one correspondence between the class of certain block tridiagonal matrices with the entries $-1, 0,$ or $1$ and the free monoid generated by $2n$ generators $\sigma _{1}, \cdots ,\sigma _{n}, \sigma _{1}^{-1},\cdots , \sigma _{n}^{-1}$ and relation $\sigma _{i}^{\pm 1}\sigma _{j}^{\pm 1} = \sigma _{j}^{\pm 1}\sigma _{i}^{\pm 1}~ (|i-j| \geq 2)$ and give some applications for braids. In particular, we give new formulation of the reduced Alexander matrices for closed braids.