Nonhomogeneous matrix equations of the form
 |
(1)
|
can be solved by taking the matrix inverse to obtain
 |
(2)
|
This equation will have a nontrivial solution iff the determinant 
. In general, more numerically stable techniques of
solving the equation include Gaussian elimination,
LU decomposition, or the square
root method.
For a homogeneous 
matrix equation
![[a_(11) a_(12) ... a_(1n); a_(21) a_(22) ... a_(2n); | | ... |; a_(n1) a_(n2) ... a_(nn)][x_1; x_2; |; x_n]=[0; 0; |; 0]](/images/equations/MatrixEquation/NumberedEquation3.svg) |
(3)
|
to be solved for the 
s,
consider the determinant
 |
(4)
|
Now multiply by 
,
which is equivalent to multiplying the first column (or any column) by 
,
 |
(5)
|
The value of the determinant is unchanged if multiples of columns are added to other columns. So add 
times column 2, ..., and 
times column 
to the first column to obtain
 |
(6)
|
But from the original matrix, each of the entries in the first columns is zero since
 |
(7)
|
so
 |
(8)
|
Therefore, if there is an 
which is a solution, the determinant
is zero. This is also true for 
, ..., 
, so the original homogeneous system has a nontrivial solution
for all 
s
only if the determinant is 0. This approach is the
basis for Cramer's rule.
Given a numerical solution to a matrix equation, the solution can be iteratively improved using the following technique. Assume that the numerically obtained solution to
 |
(9)
|
is 
,
where 
is an error term. The first solution therefore gives
 |
(10)
|
 |
(11)
|
where 
is found by solving (10)
 |
(12)
|
Combining (11) and (12) then gives
 |
(13)
|
