Since, A is a 4x5 Matrix
Therefore, it means that A has 4 Linear Equations with 5 variables.
Now, you can write the x in Ax = 0 as:
x = | |
This means that x has 2 Linear Independent variables.
This will only happen when A has 2 Free Variables.
It's a fact that,
Total number of Variables = Number of Pivot Variables + Number of Free Variables
5 = Number of Pivot Variables + 2
Number of Pivot Variables (or Rank of A) = 3