row and third column, its minor M is given by M = = – = – (obtained by deleting R and C in ∆ ). Definition Cofactor of an element a ij , denoted by A ij is defined by A ij = (– ) i + j M ij , where M ij is minor of a ij . Example Find minors and cofactors of all the elements of the determinant – Solution Minor of the element a ij is M ij Here a = . So M = Minor of a = M = Minor of the element a = M = Minor of the element a = – M = Minor of the element a = Now, cofactor of a ij is A ij .
So A = (– ) + M = (– ) ( ) = A = (– ) + M = (– ) ( ) = – A = (– ) + M = (– ) (– ) = A = (– ) + M = (– ) ( ) = DETERMINANTS Example Find minors and cofactors of the elements a , a in the determinant ∆ = Solution By definition of minors and cofactors, we have Minor of a = M = = a a – a a Cofactor of a = A = (– ) + M = a a – a a Minor of a = M = = a a – a a Cofactor of a = A = (– ) + M = (– ) ( a a – a a ) = – a a + a a Remark Expanding the determinant ∆ , in Example , along R , we have