The adjoint determinant will be seen subsequently to present itself in the theory of linear equations and in the theory of linear transformation.
Let there be 2n equations r }}?
Linear Equations.-It is of importance to study the application of the theory of determinants to the solution of a system of linear equations.
Suppose given the n equations fl= = allxl +a12x2 + ï¿½ ï¿½ ï¿½ + annxn = 0, f2 =a21x1+a22x2+ï¿½ï¿½ï¿½+a2nxn =0, fn =anlxl +an2x2+ï¿½ï¿½ï¿½ +annxn = 0.
Denote by A the determinant (a11a22ï¿½ï¿½ï¿½ann)ï¿½ Multiplying the equations by the minors A l, .., A2,,,,ï¿½ï¿½ï¿½Ani., respectively, and adding, we obtain x 1 (ai, Aig+a2p.A2lc+ï¿½ï¿½ï¿½+anï¿½Anï¿½) =xï¿½A=o, since from results already given the remaining coefficients of x 11' x 2, ...x ï¿½ 'i xï¿½+I,...x, vanish identically.