Fact checking about formula of determinant

Question

On my book, determinant is calculated this way: $det(A)= \sum_{j=1}^n (-1)^{i+j} A_{Ij} det(\hat A_{ij})$

Since we can do either column or row, can I switch $det(A)= \sum_{i=1}^n (-1)^{i+j} A_{Ij} det(\hat A_{ij})$? Fact checking.