Say you had a three by three matrix, $A$, and for $A$ one of the variables was unknown, say $α$. Using $NS(A)$, how can you determine for what values of $α$ the matrix is/ is not invertible, without touching the determinants of the matrix?
A = \begin{pmatrix} x_1 & x_2 & x_3\\ x_4&x_5&α\\ x_6&x_7&x_7 \end{pmatrix}
Have the variables as any numbers you like. How does the null space of a matrix determine anything to do with the invers-ibility of the matrix?
When you have a non trivial Nullspace, you will have some vectors with Eigenvalue of 0. Therefore your matrix will not be invertible. How the calculation of a Nullspace is easier/harder than the determinant of a $3 \times 3$ matrix is debatable but certainly, when it comes to higher dimensions.