I have to prove the following statement, or provide a counterexample:
If the coefficient matrix of a system of m linear equations in n unknowns has rank m, then the system has a solution.
My thought is that this statement is true, and I think the proof would use the following theorem:
Let Ax=b be a system of linear equations. Then the system is consistent (the solution set is nonempty) if and only if rank(A) = rank(A|b).
Is it true that if an mxn matrix has rank m as assumed that the rank of the corresponding augmented matrix (A|b) would ALWAYS have the same rank?