In numerical calculations of fluid flows and heat transfer it is often necessary to solve a system of algebraic equations with a nine-diagonal coefficient matrix. Two examples are the diffusion and pressure-correction equations when discretized on nonorthogonal grids. A method of solving such systems of equations, based on the strongly implicit procedure of Stone [1] for five-diagonal matrices, is presented. It operates on the upper and lower triangular matrices with only seven nonzero diagonals, thus requiring less storage and computing time per iteration than the alternative extensions of the strongly implicit procedure to nine-diagonal coefficient matrices. It is also more efficient than the alternative methods—for the land of equations studied—when the missing diagonals in the upper and lower triangular matrices correspond to points lying in “sharp” corners of a computational molecule. Results of various test calculations and comparisons of performance with alternative solvers are presented to support this view. The proposed solver can also be applied to five-diagonal matrix problems, in which case it reduces to the strongly implicit procedure of Stone [1].