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}](c3_752.gif) |
(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}](c3_753.gif) |
(2) |
Now multiply
by
, 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}](c3_754.gif) |
(3) |
Another property of Determinants enables us to add a constant times any column to any column and
obtain the same Determinant, so add
times column 2 and
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}](c3_755.gif) |
(4) |
If
, then (4) reduces to
, so the system has nondegenerate solutions (i.e., solutions other than
(0, 0, 0)) only if
(in which case there is a family of solutions). If
and
, the
system has no unique solution. If instead
and
, 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}](c3_761.gif) |
(5) |
and similarly for
This procedure can be generalized to a set of
equations so, given a system of
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}](c3_764.gif) |
(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}](c3_765.gif) |
(9) |
If
, then nondegenerate solutions exist only if
.
If
and
, 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}](c3_766.gif) |
(10) |
Then
for
. 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}](c3_769.gif) |
(11) |
See also Determinant, Linear Algebra, Matrix, System of Equations, Vector
© 1996-9 Eric W. Weisstein
1999-05-25