A new type of matrices which is called the first and the last difference r-block circulant matrix(bFLDCM_r) was presented. The linear operation and the matrix product were verified to bFLDCM_r. Two discriminance, a sufficient and necessary condition for determining the nonsingularity and the diagonalization of bFLDCM_r were discussed. Finally, by using Sherman-Morrison-Woodbury formula, the direct algorithm for solving r-circulant tridiagonal block systems of linear equations was provided.