So when we have the standard forms of a linear equation system and then we represent it as an augmented matrix, the constant is not included as part of the dimension of the augmented matrix, right?
So, let's say we have two variables (x and y) and three equations. This would be a 3 * 2 matrix? It would not be a 3 * 3 correct? The constants column does not count?
No, that is not correct. For example, consider the system of equations $$ x + 2y = 3,\\ 4x + 5y = 6,\\ 7x + 8y = 9. $$ The matrix equation corresponding to this system is $Ax = b$, where $$ A = \pmatrix{1 & 2\\4&5\\7&8}, \quad b = \pmatrix{3\\6\\9}. $$ The augmented matrix associated with this system is $$ M = [A \mid b] = \left[\begin{array}{cc|c} 1&2&3\\4&5&6\\ 7&8&9 \end{array} \right]. $$ The size of the augmented matrix is $3 \times 3$. The coefficient matrix $A$ has size $3 \times 2$.