Determinants and Systems of Linear Equations

Abstract

The chapter develops a procedure for the study of systems of linear equations, also useful in combination with reduction methods and known as the method of the determinants. Preliminarily, addition and multiplication of matrices, inverse and transpose matrices are defined. The ranks of the complete and associate matrices of a system of linear equation can also be calculated by means of determinants of submatrices, as stated by Kronecker’s theorem, in order to apply the theorem of Rouché-Capelli to lay down the compatibility. At this stage a procedure to ascertain the compatibility of the system of linear equation is implemented only in terms of determinants. The solutions are also found in terms of determinants by Cramer’s rule.