How to select free variables in linear equations?

Let AX = 0
A is transformed into a simplified ladder matrix by elementary row transformation
1 2 0 3 4
0 0 1 5 6
0 0 0 0 0
0 0 0 0 0
Then the column of the first non-zero element in the non-zero row corresponds to the constraint variables, in example, x1, X3
The other variables are free variables, such as X2, X4 and X5

Based on which free variables are selected for the linear basic solution system?
For example, which matrix of order 3 should I choose as a free variable? Why?
two hundred and twelve
000
000

Select 1 and 2 as free variables, mark the first non-zero number of each row first, and remove the columns of the marked numbers, and other columns are the free variables

Linear algebra, the solution of non-homogeneous equations
For the same matrix A with respect to non-homogeneous linear equations AX = B (B is not equal to 0) and homogeneous linear equations AX = 0, then ()
A. When AX = 0 has no nonzero solution, ax = B has no solution
B. When AX = 0 has infinite solutions, ax = B has infinite solutions
C. When AX = B has no solution, ax = 0 has no nonzero solution
D. When AX = B has a unique solution, ax = 0 has only a zero solution
You should choose D, but why isn't b right?
***Can a system of homogeneous linear equations have only two solutions, a zero solution and a non-zero unique solution?

When AX = 0 has no non-zero solution, then a is a full rank matrix. Then AX = B must have a solution. When AX = 0 has infinite solutions, then a must not be a full rank matrix. If AX = B has no solution and infinite Solutions R (a) ≠ R (a | B) r (a) is equal to R (a | b)

What is the basic solution system of linear algebra and how to find it

The fundamental solution is for the homogeneous linear system AX = 0
When R (a)

How to select free variables in linear equations?