: :. IV7

.............................................................................................................3 ...................................5 .............................................................................................6 ....................................................................................8 .................................................................................................13

, . . I1,I2,I3,...,In I1 I2... In , . , (S) m n , :a11x1 + a12x2 + . . . + a1nxn = b1a21x1 + a22x2 + . . . + a2nxn = b2...am1x1 + am2x2 + . . . + amnxn = bm

x1, x2, . . . , xn , ij, 1 i m, 1 j n , bi, 1 i m, . m n : m < n, m=n m > n. , . , n- (x1, x2, . . . , xn) , . . , . :2x - y = 3 4x -2 y = 1 (x,y) . (). (), (b1=b2=...=bn=0) , ; (, ) , . :x +2 y = 3 x -2 y = 1 (2,), . . (. ), . :2x - y = 1 4x -2 y = 2 (x,y) . x x = (- ), y =2 -1. (, 2 -1) , . , , n- . . :1. 2. c 03. .

: . , . :x = 1y = 2z = 0 (1,2,0), :x1+x2-x3=2x2+x3=1x3=5 . (3,-4,5) . , :a) ( )b) ( )c)

, , . =ad-bc =a-b+c :a) .b) ( ) .c) () .d) () , .e) i- () i- () i- () , .f) () () .g) () (-1)i+j, i j , , , ( ). . :[footnoteRef:2] [2: www.matematiranje.com]

,, . , , (aei) - ,,+. (gec) - -". 1[footnoteRef:3] [3: , . , .: 5 ,,1999., .69. ]

a) ;b) ( ).:a) = 2b) 1. 2. , , 5 .

XVII (Leibniz, 1646-1716.) (L'ospital, 1661-1704.). XVIII (Cramer 1704-1752.) n . .12

:

a1x + b1y = c1 a1x + b1y + c1z = d1

a2x + b2y = c2 a2x + b2y + c2z = d2 a3x + b3y + c3z = d3 :D=, Dx=, Dy=, :D= , Dx=, Dy=, Dz=.

D , n Dk, k= 1,2..., n ( Dk Dx,, Dy, Dz). Dk k- , :

, . n n . x1 = 1 ; x2 = ; . . . ; xn = n k D = Dk k = 1, 2, . . . , n. D k, k-

1, k- , 2, k- n- n k- ,

1, 2,... n ,

.

, , 1, 2, . . . n D0,

k = D0 , , xk = k=1,2,...,nAo je D=0, Dk 0, k , . , D=0 Dk 0 k, Dk = k D = 0 Dk 0. D=0 Dk=0, k=1,2,...,n , : ( ) ( ) ., , b1 = b2 = . . . = bn = 0, D1 = D2 = . . . = Dn = 0 , . , , D0 , , D=0 . . 2[footnoteRef:4] [4: , . , .: 5 ,,1999., .75.]

:2x - y - z = 43x + 4y 2z = 113x 2y + 4z = 11

D = Dx = Dy = Dz = x = =3 , y = =1 , z = =1, (3,1,1). 3[footnoteRef:5] [5: , . , .: 5 ,,1999., .76]

m n : x y + z = 1(m + 1)x + (m2 1)y + z = -1(m2 + 2)x + (m 2)y + 2z = 4

D = .Dx= Dy= .Dz= m 0 m 1 m -1, D0. x = , y = , z = , ().a m=0, D = Dx,= Dy= Dz=0. . m=0 -1 , 0 = -2. . m=1, D =0, Dx,= -40. . m = -1, D = Dx,= Dy= Dz=0. . m = -1 x y + z =1z = -13x - 3y + 2z = 4A -3 1 , 0=0, z = -1, y= , x= + 2, o ( + 2, , -1), R. .:

, . , .(1999.) ,, 5 ,- , . www.matematiranje.com , .(1979). ,, , - ,. (1994.) ,, 3. , ,