On the minimal coset coverings of the set of singular and of the set of nonsingular matrices

It is determined minimum number of cosets over linear subspaces in $F_q$ necessary to cover following two sets of $A(n\times n)$ matrices. For one of the set of matrices $\det(A)=0$ and for the other set$\det(A)\neq 0$. It is proved that for singular matrices this number is equal to $1+q+q^2+\ldots+q^{n-1}$ and for the nonsingular matrices it is equal to $\dfrac{(q^n-1)(q^n-q)(q^n-q^2)\cdots(q^n-q^{n-1})}{q^{\binom{n}{2}}}$.