Is it true that there is an interesting entry in each row of a matrix with nonzero determinant?

By doing cofactor expansion, we get that for each individual entry, the determinant is an affine linear function wet that entry. The slope is the minor at that entry.

If the entry is not interesting then the slope must be zero. If all minors of a row are zero, then by doing cofactor expansion on that row, we get that the determinant is zero.