Rank of a matrix related to consistency

Question

I have a question about the rank of matrices and its relation to a linear system being consistent. I do not intuitively understand why the rank of the coefficient matrix being equal to the rank of the augmented matrix grants consistency for a linear system of equations. I understand that linear systems of equations that are consistent have this property, but why is that the case?

Answer 1

To solve a linear system, we perform the elimination on it. After a series of routine operations, we obtain those equations that are truly efficient to provide the solution; meanwhile the LHS of the rest of them are cancelled to be 0. If the ranks agree, then we will not obtain some equation like $0 = 1$ which is obviously false. Otherwise $0=1$ holds for no solutions of this linear system, hence inconsistent.

In a textbook for a first course of linear algebra, it might be proved that after Gaussian elimination, the linear system could be equivalently convert to something like this [maybe 5 equations in 8 unknowns]:

\begin{alignat*}{9} &x_1& + & 0 x_2 &+& a_3 x_3 & + &0x_4& + &a_5 x_5& + &a_6x_6& + &0x_7& + &0x_8& = &y_a&\\ & & & 1x_2 & +& b_3 x_3 & + &0x_4 & +& b_5x_5& + &b_6x_6& + &0x_7& + &0x_8& = &y_b&\\ & & & & & & &1x_4& +& c_5 x_5& + &c_6 x_6& + &0x_7& + &0x_8& = &y_c&\\ & & & & && && && && &1x_7& + &0x_8& = &y_d&\\ && && && && && && && &1x_8& =&y_e& \end{alignat*} Clearly this is consistent no matter which values $a_j, b_j$ etc. take. This is correspondent to the so called "row-reduced echelon form".

Answer 2

Rank is the maximal number of linearly independent columns. If the rank of the augmented matrix $[A|b]$ is equal to the rank of the coefficient matrix $A$, this means that the right-hand side column $b$ is a linear combination of the columns of $A$ (otherwise the rank of the augmented matrix is higher). And that is exactly the existence of a solution: $Ax$ is a linear combination of columns of $A$ with coefficients given by the coordinates of $x$, so a solution of $Ax=b$ exists if an only if $b$ is a linear combination of columns of $A$.