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The Adjoint or Reciprocal Determinant arises from A = (a11a22a33 ...a nn) by substituting for each element A ik the corresponding minor Aik so as to form D = (A 11 A 22 A 33 ï¿½ï¿½ï¿½ A nn).

00Its value is therefore O n and we have the identity D.0 = A n or D It can now be proved that the first minor of the adjoint determinant, say B rs is equal to An-2aï¿½.

00The adjoint determinant is the (n - I) th power of the original determinant.

00The adjoint determinant will be seen subsequently to present itself in the theory of linear equations and in the theory of linear transformation.

00It is easy to see that the adjoint determinant is also 'symmetrical, viz.

00For the second order we may take Ob - I - A, 1 1 +A2, and the adjoint determinant is the same; hence (1 +A2)x1 = (1-A 2)X 1 + 2AX2, (l +A 2)x 2 = - 2AX1 +(1 - A2)X2.

00Similarly, for the order 3, we take 1 v Ab= -v 1 A =1 +x2 + 1, 2 + ï¿½ - A 1 and the adjoint is 1+A v +Aï¿½ -ï¿½ +Av -v +Aï¿½ 1+11 2 A +/-tv pt+AvA +ï¿½v 1 +1,2 leading to the orthogonal substitution Abx1 = (1 +A 2 - / 22 - v 2) X l +2(v+Aï¿½)X2 +2(/1 +Av)X3 1bx2 = 2(Aï¿½ - v)Xl+(1 +ï¿½2 - A2 - v2)X2 / +2(Fiv+A)X3 Abx3 = 2(Av +ï¿½)X1 +2(/lv-A)X2+(1+v2-A2- (12)X3.

00The Adjoint or Reciprocal Determinant arises from A = (a11a22a33 ...a nn) by substituting for each element A ik the corresponding minor Aik so as to form D = (A 11 A 22 A 33 Ã¯¿½Ã¯¿½Ã¯¿½ A nn).

00For the second order we may take Ob - I - A, 1 1 +A2, and the adjoint determinant is the same; hence (1 +A2)x1 = (1-A 2)X 1 + 2AX2, (l +A 2)x 2 = - 2AX1 +(1 - A2)X2.

00Similarly, for the order 3, we take 1 v Ab= -v 1 A =1 +x2 + 1, 2 + Ã¯¿½ - A 1 and the adjoint is 1+A v +AÃ¯¿½ -Ã¯¿½ +Av -v +AÃ¯¿½ 1+11 2 A +/-tv pt+AvA +Ã¯¿½v 1 +1,2 leading to the orthogonal substitution Abx1 = (1 +A 2 - / 22 - v 2) X l +2(v+AÃ¯¿½)X2 +2(/1 +Av)X3 1bx2 = 2(AÃ¯¿½ - v)Xl+(1 +Ã¯¿½2 - A2 - v2)X2 / +2(Fiv+A)X3 Abx3 = 2(Av +Ã¯¿½)X1 +2(/lv-A)X2+(1+v2-A2- (12)X3.

00In 1768, recognized as a man who had both the ability and the means for a scientific career, he was nominated adjoint chimiste to the Academy, and in that capacity made numerous reports on the most diverse subjects, from the theory of colours to water-supply and from invalid chairs to mesmerism and the divining rod.

00The same year he obtained the position of adjoint to Baudon, one of the farmers-general of the revenue, subsequently becoming a full titular member of the body.

00Higher ranks would use stars to denote their status, starting with one silver star for the Chief Regional Adjoint and four gold stars on the epaulet for the Delegue General de la Malice en Zone Nord.

00The same year he obtained the position of adjoint to Baudon, one of the farmers-general of the revenue, subsequently becoming a full titular member of the body.

01Its value is therefore O n and we have the identity D.0 = A n or D It can now be proved that the first minor of the adjoint determinant, say B rs is equal to An-2aÃ¯¿½.

01From the equations a11xi+ a12x2+ a13x3 +Ã¯¿½Ã¯¿½Ã¯¿½ = El, a21x1+a72x2+ a23x3+Ã¯¿½Ã¯¿½Ã¯¿½ = 2, a3lxl+a32x2+a33x3+Ã¯¿½Ã¯¿½Ã¯¿½ = 53, 0x1 =A111+A21E2+A31Er3+Ã¯¿½Ã¯¿½Ã¯¿½ 0x2 = Al2E1 + A22E2+ A32Srr3+Ã¯¿½Ã¯¿½Ã¯¿½ AX3 =A13E1+A23E2+A33E3+Ã¯¿½Ã¯¿½Ã¯¿½ A n 1 E1 = B110x1 + B12Ax2+ B13Ax3+Ã¯¿½Ã¯¿½Ã¯¿½, On - lt2 = B 21Ax1+ B220x2+ B230x3+Ã¯¿½Ã¯¿½Ã¯¿½ An-15513 = B31Ax1 + B 32Ax2+B330x3+Ã¯¿½Ã¯¿½Ã¯¿½ and comparison of the first and third systems yields B = An-2a rs = rsÃ¯¿½ In general it can be proved that any minor of order of the adjoint is equal to the complementary of the corresponding minor of the original multiplied by the h power of the original determinant.

01The adjoint determinant is the (n - I) th power of the original determinant.

01The adjoint determinant will be seen subsequently to present itself in the theory of linear equations and in the theory of linear transformation.

01It is easy to see that the adjoint determinant is also 'symmetrical, viz.

01

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