ALGEBRA. 



1.352 If dij = dji, and an = o, the determinant is a skew symmetrical 

 determinant. 



A skew symmetrical determinant of even order is a perfect squa're. 



A skew symmetrical determinant of odd order vanishes. 



1.360 A system of linear equations: 



+ di2X2 + + di n X n = fa 



= fa 



= k n 



has a solution: 



provided that 



A = | an | * o. 



1.361 If A = o, but all the first minors are not o, 



<J^ d 2 A 



^\ / v* ^ / v* /\ ' I ^^^ * 1? ( 7 T o ^7, | 



^^^ dd ss d(lrj 



where 5 may be any one of the integers i, 2, . . . . , n. 



1.362 If fa = fa = . . . o . . = k n = o, the linear equations are homogeneous, 

 and if A = o, 



Xj X s ,._ v 



1.363 The condition that n linear homogeneous equations in n variables shall 

 be consistent is that the determinant, A, shall vanish. 



1.364 If there are n + i linear equations in n variables : 



+ #12#2 + + dinX n = fa 



+ d22X2 + + d2nX n = fa 



C\Xi -\- 2X2 ~T~ ~\C n Xn = ^n+1 



the condition that this system shall be consistent is that the determinant' 



di2 din . fa =0 



#22 ^2" fa 



C2 Cn k n +l 



