克莱姆法则
若n元线性方程组
{ a 11 x 1 + a 12 x 2 + . . . a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + . . . a 2 n x n = b 2 . . . a n 1 x 1 + a n 2 x 2 + . . . a n n x n = b n \begin{cases} a_{11}x_1+a_{12}x_2+...a_{1n}x_n=b_1\\ a_{21}x_1+a_{22}x_2+...a_{2n}x_n=b_2\\ ...\\ a_{n1}x_1+a_{n2}x_2+...a_{nn}x_n=b_n\\ \end{cases} ⎩⎪⎪⎪⎨⎪⎪⎪⎧a11x1+a12x2+...a1nxn=b1a21x1+a22x2+...a2nxn=b2...an1x1+an2x2+...annxn=bn
的系数行列式不等于0，即：
D = ∣ a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n . . . . . . . . . . . . a n 1 a n 2 . . . a n n ∣ ≠ 0 D= \begin{vmatrix} a_{11} & a_{12} & ... & a_{1n}\\ a_{21} & a_{22} & ... & a_{2n}\\ ... & ... & ... & ...\\ a_{n1} & a_{n2} & ... & a_{nn}\\ \end{vmatrix} \neq 0 D=∣∣∣∣∣∣∣∣a11a21...an1a12a22...an2............a1na2n...ann∣∣∣∣∣∣∣∤=0
则方程组有唯一解，且
x 1 = D 1 D , x 2 = D 2 D , . . . , x n = D n D x_1 = \frac{D_1}{D},x_2 = \frac{D_2}{D},...,x_n = \frac{D_n}{D} x1=DD1,x2=DD2,...,xn=DDn
其中 D j ( j = 1 , 2 , 3 , . . , n ) D_j(j = 1,2,3,..,n) Dj(j=1,2,3,..,n)是将系数行列式中第 j j j列用常数项 b 1 , b 2 , . . . , b n b_1,b_2,...,b_n b1,b2,...,bn代替后得到的 n n n阶行列式。
D = ∣ a 11 . . . a 1 , j − 1 b 1 a 1 , j + 1 . . . a 1 n a 21 . . . a 2 , j − 1 b 2 a 2 , j + 1 . . . a 2 n . . . . . . . . . . . . . . . . . . . . . a n 1 . . . a n , j − 1 b n a n , j + 1 . . . a n n ∣ D= \begin{vmatrix} a_{11} & ... & a_{1,j-1} & b_1 & a_{1,j+1} & ... & a_{1n} \\ a_{21} & ... & a_{2,j-1} & b_2 & a_{2,j+1} & ... & a_{2n} \\ ... & ... & ... & ... & ... & ... & ...\\ a_{n1} & ... & a_{n,j-1} & b_n & a_{n,j+1} & ... & a_{nn} \\ \end{vmatrix} D=∣∣∣∣∣∣∣∣a11a21...an1............a1,j−1a2,j−1...an,j−1b1b2...bna1,j+1a2,j+1...an,j+1............a1na2n...ann∣∣∣∣∣∣∣∣
当方程组右边的常数 b j b_j bj不全为零时，方程组称为非齐次线性方程组；当 b 1 b_1 b1 = b 2 b_2 b2 = … b n b_n bn = 0时，方程组称为齐次线性方程组。