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Cramer's Rule

Given a set of linear equations

\begin{displaymath}
\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}
\end{displaymath} (1)

consider the Determinant
\begin{displaymath}
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.
\end{displaymath} (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
\begin{displaymath}
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.
\end{displaymath} (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,
\begin{displaymath}
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.
\end{displaymath} (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
\begin{displaymath}
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},
\end{displaymath} (5)

and similarly for
$\displaystyle y$ $\textstyle =$ $\displaystyle {\left\vert\begin{array}{ccc}a_1 & d_1 & c_1\\  a_2 & d_2 & c_2\\  a_3 & d_3 & c_3\end{array}\right\vert \over D}$ (6)
$\displaystyle z$ $\textstyle =$ $\displaystyle {\left\vert\begin{array}{ccc}a_1 & b_1 & d_1\\  a_2 & b_2 & d_2\\  a_3 & b_3 & d_3\end{array}\right\vert \over D}.$ (7)


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

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

let
\begin{displaymath}
D \equiv \left\vert\matrix{
a_{11} & a_{12} & \cdots & a_{1...
... \vdots\cr
a_{1n1} & a_{n2} & \cdots & a_{nn}\cr}\right\vert.
\end{displaymath} (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
\begin{displaymath}
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.
\end{displaymath} (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
\begin{displaymath}
({\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}.
\end{displaymath} (11)

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



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© 1996-9 Eric W. Weisstein
1999-05-25