Explicit formula for the inverse of a tridiagonal matrix by backward continued fractions

In this paper, we consider a general tridiagonal matrix and give the explicit formula for the elements of its inverse. For this purpose, considering usual continued fraction, we define backward continued fraction for a real number and give some basic results on backward continued fraction. We give the relationships between the usual and backward continued fractions. Then we reobtain the LU factorization and determinant of a tridiagonal matrix. Furthermore, we give an efficient and fast computing method to obtain the elements of the inverse of a tridiagonal matrix by backward continued fractions. Comparing the earlier result and our result on the elements of the inverse of a tridiagonal matrix, it is seen that our method is more convenient, efficient and fast.