Let P = [pi,j ]i,j≥0 be an infinite matrix whose entries satisfy pi,j = μpi,j−1 + λpi−1,j + νpi−1,j−1 for i, j ≥ 1, and whose first column resp. row satisfy linear recurrences with constant coefficients of orders ρ resp. σ. Then we show that its principal minors dn satisfy dn = ∑δ j=1 cjω dn−j where cj are constants, ω = λμ + ν, and δ = (ρ+σ−2 ρ−1 ) . This implies a recent conjecture of Bacher .