Cramer's Rule

Given a set of linear equations

$$\cases{a_1x+b_1y+c_1z=d_1\cr a_2x+b_2y+c_2z=d_2\cr a_3x+b_3y+c_3z=d_3,\cr}$$ (1)

consider the Determinant

$$D\equiv \left\vert\matrix{a_1 & b_1 & c_1\cr a_2 & b_2 & c_2\cr a_3 & b_3 & c_3\cr}\right\vert.$$ (2)

Now multiply $D$ by $x$, and use the property of Determinants that Multiplication by a constant is equivalent to Multiplication of each entry in a given row by that constant

$$x \left\vert\matrix{a_1 & b_1 & c_1\cr a_2 & b_2 & c_2\cr a... ...& c_1\cr a_2x & b_2 & c_2\cr a_3x & b_3 & c_3\cr}\right\vert.$$ (3)

Another property of Determinants enables us to add a constant times any column to any column and obtain the same Determinant, so add $y$ times column 2 and $z$ times column 3 to column 1,

$$x D = \left\vert\matrix{a_1x+b_1y+c_1z & b_1 & c_1\cr a_2x+b... ..._1 & c_1\cr d_2 & b_2 & c_2\cr d_3 & b_3 & c_3\cr}\right\vert.$$ (4)

If ${\bf d}={\bf0}$, then (4) reduces to $xD=0$, so the system has nondegenerate solutions (i.e., solutions other than (0, 0, 0)) only if $D=0$ (in which case there is a family of solutions). If ${\bf d}\not={\bf0}$ and $D=0$, the system has no unique solution. If instead ${\bf d}\not={\bf0}$ and $D\not=0$, then solutions are given by

$$x = {\left\vert\matrix{d_1 & b_1 & c_1\cr d_2 & b_2 & c_2\cr d_3 & b_3 & c_3\cr}\right\vert \over D},$$ (5)

and similarly for

$$y = {\left\vert\begin{array}{ccc}a_1 & d_1 & c_1\\ a_2 & d_2 & c_2\\ a_3 & d_3 & c_3\end{array}\right\vert \over D}$$ (6)

$$z = {\left\vert\begin{array}{ccc}a_1 & b_1 & d_1\\ a_2 & b_2 & d_2\\ a_3 & b_3 & d_3\end{array}\right\vert \over D}.$$ (7)

This procedure can be generalized to a set of $n$ equations so, given a system of $n$ linear equations

$$\left[{\matrix{ a_{11} & a_{12} & \cdots & a_{1n}\cr \vdot... ...r}}\right] = \left[{\matrix{d_1\cr \vdots\cr d_n\cr}}\right],$$ (8)

let

$$D \equiv \left\vert\matrix{ a_{11} & a_{12} & \cdots & a_{1... ... \vdots\cr a_{1n1} & a_{n2} & \cdots & a_{nn}\cr}\right\vert.$$ (9)

If ${\bf d}={\bf0}$, then nondegenerate solutions exist only if $D=0$. If ${\bf d}\not={\bf0}$ and $D=0$, the system has no unique solution. Otherwise, compute

$$D_k \equiv \left\vert\matrix{ a_{11} & \cdots & a_{1(k-1)} ... ...{n(k-1)} & d_n & a_{n(k+1)} & \cdots & a_{nn}\cr}\right\vert.$$ (10)

Then $x_k = {D_k/D}$ for $1 \leq k \leq n$. In the 3-D case, the Vector analog of Cramer's rule is

$$({\bf A}\times{\bf B})\times({\bf C}\times {\bf D}) = ({\bf... ...imes{\bf D}){\bf C}-({\bf A}\cdot{\bf B}\times{\bf C}){\bf D}.$$ (11)

See also Determinant, Linear Algebra, Matrix, System of Equations, Vector