Example 1: Use of matrices in solving linear systems of equations
The
coefficients
of the linear system of equations
(1)
x
1
+ 3x
2
- x
3
= 1
x
1
- 2x
2
+ x
3
= 0
2x
1
- x
2
+ x
3
= 1
form a matrix
A =
1
3
-1
,
1
-2
1
2
-1
1
which we use (together with a suitable definition of matrix multiplication) to rewrite (1) as
1
3
-1
1
-2
1
2
-1
1
x
1
x
2
x
3
=
1
0
1
i.e.,
A
=
.
Moreover, later we shall see that (1) can be written in an equivalent form
x
1
+ 3x
2
-x
3
= 1
or
- 5x
2
+ 2x
3
= -1
x
3
= 1
1
3
-1
0
-5
1
0
0
1
x
1
x
2
x
3
=
1
,
-1
2
and this system can easily be solved by the
substitution method
.
Go back
= 
.