对于一个三乘三的系统
a x + b y + c z = j d x + e y + f z = k g x + h y + i z = l {\displaystyle {\begin{array}{c}ax+by+cz=j\\dx+ey+fz=k\\gx+hy+iz=l\\\end{array}}} 以矩阵形式 [ a b c d e f g h i ] [ x y z ] = [ j k l ] {\displaystyle {\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}}{\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}j\\k\\l\end{bmatrix}}}
x = | j b c k e f l h i | | a b c d e f g h i | {\displaystyle x={\frac {\begin{vmatrix}j&b&c\\k&e&f\\l&h&i\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}}} y = | a j c d k f g l i | | a b c d e f g h i | {\displaystyle y={\frac {\begin{vmatrix}a&j&c\\d&k&f\\g&l&i\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}}} z = | a b j d e k g h l | | a b c d e f g h i | {\displaystyle z={\frac {\begin{vmatrix}a&b&j\\d&e&k\\g&h&l\end{vmatrix}}{\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}}}
一般来说： A x = b {\displaystyle Ax=b} x i = | A i | | A | ∀ i = { 1 , … , n } {\displaystyle x_{i}={\frac {|A_{i}|}{|A|}}\;\forall i=\{1,\dots ,n\}} 其中 A i {\displaystyle A_{i}} 是通过用向量 b 替换与 x i {\displaystyle x_{i}} 相关的向量而形成的。
