Matrix factorizations such as LU, Cholesky and others are widely used for solving linear systems. In particular, the diagonal pivoting method can be applied to symmetric and indefinite matrices. Floating-point arithmetic is extensively used for this purpose. Since finite precision numbers are treated, rounding errors are involved in computed results. In this paper rigorous backward error bounds for 2×2 linear systems which arise in the factorization process of the diagonal pivoting method are given. These bounds are much better than previously known ones.