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.
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2"Interesting" question. But I don't understand the sentence "I know that each entry of every matrix with nonzero determinant is not interesting" – Jean Marie Jun 22 '20 at 18:47
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1It is my opinion that it is not possible for each of single entry in matrix can make determinant 0 – Subu Jun 22 '20 at 21:28
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Yes someone answered – Subu Jun 23 '20 at 13:38
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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.
David Lui
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[+1] but the "slope" as you call it is strictly speaking the cofactor, not the minor. – Jean Marie Jun 22 '20 at 19:07
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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.
subhanceo
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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.
J Cameron
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You don't need to delete the zero terms, but only the nonzero terms are going to give interesting entries of the row since if the determinant of the minor $X_j$ is zero changing $a_j$ won't change the determinant. – J Cameron Jun 23 '20 at 17:22