The article describes one of the methods for computing determinants without using fractions proposed by Bareiss. This problem has a clear algorithmic character in nature and refers to the field of computer algebra. The implementation of this algorithm is proposed in the known Maxima system of symbolic computations. In addition, this method makes it possible to get enough convenient formula for the calculation of the matrix of unitriangular transformation of a quadratic form to a canonical one.

The possibility of producing 3-aminobenzyl alcohol and 3-aminobenzaldehyde by oxidation of 3-aminotoluene with ozone in the solution of acetic anhydrite in the presence of manganese (II) acetate, potassium bromide and sulfuric acid has been shown. The catalytic systems for regulating selectivity and depth of substrate oxidation has been developed. The catalytic system Mn(OAc)₂ – Ac₂0 – H₂SO₄ promotes the formation of alcohol (65.5%) and 3- acetylaminobenzylidendiacetate (20.1%) with the system Mn(OAc)₂ – KBr – Ac₂O – H2SO₄ increases oxidation selectivity on the methyl group to 90.8% producing mainly aldehyde (80.8%) The optimum temperature of selective oxidation of 3– aminotoluene with the ozone – air mixture (30°C) which is much lower than that of oxidation by the known methods (120°-240°C) and the optimum rations of the reagents concentrations: for alcohol synthesis – [ArCH3] : [Mn(OAc)₂] : [H₂SO₄] =1 : 0.2 : 2.5; for aldehyde synthesis –[ ArCH3]: [Mn(OAc)₂] : [KBr] aldehyde synthesis – [ArCH3] : [Mn(OAc)₂] : [KBr] : [H₂SO₄] = 1:0.2:2.5 have been determined.