Is it true that there is an interesting entry in each row of a matrix with nonzero determinant? We call an entry of an $ n × n $ matrix with nonzero determinant interesting, if by changing this entry
(and only this) the determinant of the matrix can be made $0$.
Is it true that there is an interesting entry in each row of a matrix with nonzero determinant?
I know that each entry of every matrix with nonzero determinant is not interesting. However I could not find proof if this is a case in each row.
 A: 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.
A: Let us consider an entry $a$ of a matrix $M$; if its cofactor (or minor) is non zero, it is automatically "interesting" because the determinant of $M$ is a first degree polynomial $ua+v$ in this entry with $u \neq 0$. It suffices then to take the value $a=-v/u.$
Your question : Yes necessarily, because if all the cofactors of a certain row are zero, the determinant itself is zero ; contradiction.
A: This is true. You know that the determinant can be obtained by expanding along any row.
So, if you want to show that a given row has an interesting entry expand the determinant along that row. This gives an equation $\sum_{i=1}^n a_i X_i=d$ where the $a_i$ are $\pm$ the entries of your row, $X_i$ is the determinant of one of the minors of your matrix, and $d$ is the determinant. Since $d$ is nonzero, some of the terms $a_iX_i$ are nonzero, so if you remove the zero terms each of the remaining $a_i$ terms is interesting.
