Let m, n be positive integers or monic polynomials in $mathbb{F}_q[T]$ with n|m. Let $K_m;and;K^+_m$ be the m-th cyclotomic field and its maximal real subfield, respectively. In this paper we define two matrices $D^+_{m,n};and;D^-_{m,n}$ whose determinants give us the ratios $frac{h(mathcal{O}_{K^+_m})}{h(mathcal{O}_{K^+_n})}$ and $frac{h-(mathcal{O}_K_m)}{h-(mathcal{O}_K_n)}$ with some factors, respectively.