By using the concept of ℐ -convergence defined by Kostyrko et al. in 2001, the ℐ -limit superior of real sequences was introduced and the inequality ℐ − lim ⁡ sup ⁡ ( A x ) ≤ ℐ − lim ⁡ sup ⁡ ( x ) for all x ∈ ℓ ∞ was studied by Demirci in 2001. In this paper, we have characterized a class of ℐ -conservative matrices by studying some new inequalities related to the ℐ -limit superior.