In Linear Algebra over finite fields, a characteristic-dependent linear rank inequality is a linear inequality that holds by ranks of spans of vector subspaces of a finite dimensional vector space over a finite field of determined characteristic, and does not in general hold over fields with other characteristic. This paper shows a preliminary result in the production of these inequalities. We produce three new inequalities in 21 variables using as guide a particular binary matrix, with entries in a finite field, whose rank is 8, with characteristic 2; 9 with characteristic 3; or 10 with characteristic neither 2 nor 3. The first inequality is true over fields whose characteristic is 2; the second inequality is true over fields whose characteristic is 2 or 3; the third inequality is true over fields whose characteristic is neither 2 nor 3.