- Use your helper application
to solve the system
| x1 |
|
|
- |
x3 |
+ |
3x4 |
= |
1 |
| 2x1 |
+ |
x2 |
+ |
4x3 |
- |
2x4 |
= |
-2 |
|
- |
5x2 |
|
|
+ |
x4 |
= |
11 |
| -x1 |
+ |
2x2 |
- |
x3 |
+ |
3x4 |
= |
3 |
Note that the matrix of coefficients
for this system is the matrix A already defined in your worksheet.
Thus, the system can be written in matrix form as Ax = b, where b
is the column vector [1,-2,11,3]T. Check the answer by multiplying
the column vector x = [x1,x2,x3,x4]T
on the left by A to see if the result is b.
- Recall that A is invertible.
Calculate b as A-1x to see if you get the same result.
Note that this system has a unique solution.
- Now solve the system
| 2x1 |
+ |
x2 |
|
|
+ |
7x4 |
= |
1 |
| -2x1 |
+ |
5x2 |
+ |
x3 |
+ |
2x4 |
= |
-2 |
| 4x1 |
+ |
x2 |
+ |
3x3 |
- |
6x4 |
= |
11 |
| 4x1 |
- |
4x2 |
- |
x3 |
+ |
5x4 |
= |
3 |
What does the answer mean? The
matrix of coefficients for this system is B, which we saw was not invertible.
This is an example of a linear system with infinitely many solutions.
- Write down one specific solution
x of the system in the preceding step. Multiply on the left by B,
and confirm that Bx really is b. Now do it again with
a different solution.
- Change the constant term in
the first equation to anything other than 1, and try to solve the resulting
system. What happens? This is an example of an inconsistent system
-- it has no solution.
