Simple question involving matrices?

Question

When you have a matrix in reduced row echelon form and you get the result.....

1 0 0 | 2
0 1 0 | 4
0 0 0 | 0

Does this mean that there are infinitely many solutions because of 0 = 0?

Please note that the last column in the matrix is the solution, i.e. Ax=b

Thanks a bunch

This one is for zig or anyone else who is bored : )

This does not reflect my previous example, however, you will get a row (0 0 0 0 0), where 0 = 0.

1 0 2 4 | -8
0 1 -3 -1 | 6
3 4 -6 8 | 0
0 -1 3 4 | 12

Answer 1

You are right. The last row represents the equation 0*x=0 which is an identity. So effectively you have 2 equations and 3 unknowns. You get unique solution for two of the unknowns and any value for the third will do and that means an infinitude of solutions.

Answer 2

Yes, it does. But more specifically, you have:

x = 2
y = 4
z = anything

So the infinitely many solutions lie on the line perpendicular to the x-y plane that "pierces" the x-y plane in the point (2, 4).

Answer 3

You are right, there are infinitely many solutions.

More specifically, the reduced row echelon form that you have above means that x_1 = 2, x_2 = 4, x_3 = anything.

Answer 4

No this means that you reduced the matrix incorrectly. If you post the original you should get lots of answers helping you. Thanks.