Non-Homogeneous Question Linear Algebra

[1LastTry]

Homework Statement

For a nonhomogeneous system of 2012 equations in 1999 unknowns, answer the following three questions:

Can the system be inconsistent?
Can the system have infinitely many solutions
Can the system have a unique solutions?

Homework Equations

The Attempt at a Solution

I did the answer, but not sure if it is correct, however my prof never did it in this way not sure if it is a valid answer.

My answer to these three sub questions are: Yes, Yes, Yes

1. reason for Inconsistency:

Since the system is nonhomogeneous (The constant matrix cannot =0)
so there could be a row (0...01 or 0=1) so it would be inconsistent.

2. reason for Infinite:

Infinite amount of solutions if number of variables > rank
So if the equations are Linearly Dependent then there is really only ONE equation
So the number of variables > rank, therefor containing infinite amount of solutions
(i am not sure of this answer)

3. Reason for Unique solution.
if some of the equations are linearly dependent then the number of ranks = number of variables therefore contains a unique solution (not sure of this answer).

Please help me varify this for me or is there a better and clearer solution? Thanks A LOOT.

 

[voko]

Those wonderful dimensions, 2012 and 1999, are only to confuse you. You can investigate the correctness of your answers by constructing systems of 3 equations and 2 variables.