This is a homogenous equation and having a non-trivial solution which means it’s solution set is obtained from solving using parametric method.
In question it’s given that matrix is having 4 rows and 5 cols and since all the solutions are scalar multiple of 1-column vector means there will be only one free variable in the P matrix.
Since, there is only 1 free variable we are left with 4 columns in P with Pivots which will be counted as a rank bcz rank of matrix = no. of cols with pivots.