What are the rank conditions for consistency of a linear algebraic system?

[cheesefondue]

what are the rank conditions for consistency of a linear algebraic system?
my proffessor said that the coefficient matrix augmented with the column value matrix must have the same rank as the coefficient matrix for consistency of the system of equations. however does the term rank apply to rectangular matrices such as an augmented matrix? I if so how is it calculated also I would like to know how to deduce this condition by reasoning [the derivation].

 

[HallsofIvy]

Well, what is your definition of "rank"? The definition I would use is that the rank of a matrix is the number of non-zero rows left after you row-reduce the matrix. Obviously, that idea applies to non-square matrices. In fact, if you append a new column to a square matrix, to form the "augmented matrix", any non-zero row, after row-reduction, for the square matrix will still be non-zero for the augmented matrix- add values on the end can't destroy non-zero values already there. The only way the rank could be changed is if you have non-zero values in the new column on a row that is all zeroes except for that, so that the augmented rank has greater rank than the original matrix. That tells you that one of your matrices has reduced to 0x+ 0y+ 0z+ ...= a where a is non-zero and that is impossible. If there is no such case, you have at least one solution to each equation. Yes, the system is consistent if and only if the rank of the coefficient matrix is the same as the rank of the augmented matrix.

 

[cheesefondue]

Well my textbook says that the rank of a matrix is the dimension of the largest square sub matrix whose determinant is non-zero. we just started the chapter so I'm not sure what you mean by row reduction, could you please elaborate? I'm sorry, I konw that these are really simple questions but I'd like to know the answers nevertheless... thanx.

 

[Defennder]

Row reduction refers to the algorithmic procedure of Gaussian elimination. There are three row-reduction techniques:

1. Swapping rows
2. Multiplying a row by a constant.
3. Adding a multiple of a row to another.

See here for details:
http://en.wikipedia.org/wiki/Elementary_row_operations#Operations