The combined dimensionality of the two smaller determinants equals the dimensionality of the full determinant. Such a factorization is advantageous in that it makes it possible to perform the calculation of the ratio $R$ and the updating of the inverse matrix separately for $\vert\hat{D}\vert_\uparrow$ and $\vert\hat{D}\vert_\downarrow$:

$$\frac{\vert\hat{D}\vert^\mathrm{new}}{\vert\hat{D}\vert^\mathrm{old}} = \frac{\vert\hat{D}\vert^\mathrm{new}_\uparrow} {\vert\hat{D}\vert^\mathrm{old}_\uparrow}\cdot \frac{\vert\hat{D}\vert^\mathrm{new}_\downarrow }{\vert\hat{D}\vert^\mathrm{old}_\downarrow}$$

This reduces the calculation time by a constant factor. The maximal time reduction happens in a system of equal numbers of $\uparrow$ and $\downarrow$ particles, so that the two factorized determinants are half the size of the original one.