  In Linear Algebra over finite fields, a characteristic-dependent linear rank
inequality is a linear inequality that holds by ranks of subspaces of a vector
space over a finite field of determined characteristic, and does not in general
hold over other characteristics. In this paper, we show a method to produce
these inequalities using binary matrices with suitable ranks over different
fields. In particular, for each $n\geq7$, we produce $2\left\lfloor
\frac{n-1}{2}\right\rfloor -4$ characteristic-dependent linear rank
inequalities over $n$ variables. Many of the inequalities obtained are new but
some of them imply the inequalities presented in [1,9].
