This paper presents an alternate quadrant interlocking factorization (A.Q.I.F.) method for the solution of the linear systems that is suitable for S.I.M.D. parallel computers. A.Q.I.E is a variant of Gaussian elimination that works from the middle outward rather than from top down. This factorization generalizes the new quadrant interlocking factorization of M. M. Chawla and K. Passi [1]. This paper includes the proofs of existence of the factorization in the symmetric positive definite and nonsingular diagonally dominant cases. Results of A.Q.I.E method arc compared with quadrant interlocking factorization (Q.I.E) method of D. J. Evans and M. Hatzopoulos [2] and with Gaussian elimination.