I need to verify the linear dependence or independence of $3 \times 2$ complex matrix, how do I compute the determinant? I would use the row reduced echelon form but I have no idea about how to do that with complex numbers, can I divide for the "complex number" when row reducing or my operations of row reduction must be limited to real numbers?
Compute the cross product of the two column vectors of $A$. The entries of the resulting triple of complex numbers are nothing else but the three $2\times2$ subdeterminants of $A$. If at least one of these entries is nonzero the matrix $A$ has rank $2$. If all three entries are zero, but $A$ is not the zero matrix, then $A$ has rank $1$.