"Can you explain why and how we can check spanning by using determinant?"

I presume you mean questions 11 and 12?

We need to check if the vectors span say, R^2. So we make a general vector, (u_1, u_2), and see if we can make it out of the spanning vectors.

To do that we solve the matrix equation (augmented with u_1, u_2).

If the determinant is non-zero then the system will have a unique solution no matter what u_1 and u_2 we choose. So we can say that the span is R^2.

If, however, the determinant is 0 then we might row reduce and find that only specific values of u_1 and u_2 have solutions. That would mean that the vectors do not span R^2, they span some subspace instead (a line for example).