Let a linear system of equations be denoted
![\begin{displaymath}
{\hbox{\sf A}}{\bf X} = {\bf Y},
\end{displaymath}](s3_1952.gif) |
(1) |
where
is a Matrix and X and Y are Vectors. As shown by Cramer's Rule,
there is a unique solution if
has a Matrix Inverse
. In this case,
![\begin{displaymath}
{\bf X}={\hbox{\sf A}}^{-1} {\bf Y}.
\end{displaymath}](s3_1954.gif) |
(2) |
If
, then the solution is
. If
has no Matrix Inverse, then the solution
Subspace is either a Line or the Empty Set. If two equations are multiples of each other, solutions
are of the form
![\begin{displaymath}
{\bf X}={\bf A}+t{\bf B},
\end{displaymath}](s3_1957.gif) |
(3) |
for
a Real Number.
See also Cramer's Rule, Matrix Inverse
© 1996-9 Eric W. Weisstein
1999-05-26