Let $M$ be a finite matroid on ground set $E$ with rank $r_M(E)>0$, and let $N=M^*$ be the dual matroid. For every positive-rank finite matroid $P$, define its beta invariant by
where $\chi_P$ is the characteristic polynomial of $P$. If $M$ is disconnected, then $\beta(M)=0$. If $M$ is connected and $|E|\geq 2$, then $\beta(M)>0$. Moreover, if $r_N(E)>0$, then $\beta(M)=\beta(N)$.