Linear system and matrices: For the system of linear equations, the matrix representation is,where
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[proof:]
: A is nonsingular,
then there exists an unique
. Thus
.
: Suppose
has a unique solution
d
. Therefore,
the augmented matrix
can be transformed to the matrix in reduced row echelon form
via elementary row operations. Thus, we can employ the same row operations
to the augmented matrix
and the resulting matrix is
.
D
is the inverse of
A
.
Motivating example (continue):
Solve for the linear system
.
[solution:]
.
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: A is nonsingular, 
. Thus
.
: Suppose 
can be transformed to the matrix in reduced row echelon form 
via elementary row operations. Thus, we can employ the same row operations 
and the resulting matrix is 
. D 
.
.