## Permutable line

We say that a line in a matrix is *permutable *if permuting its elements does not change the determinant of . Prove that if has two permutable lines then its determinant is zero.

*Romanian National Olympiad, 11th grade*

* Proof: *First, let’s see what means for a line to be permutable. For the line , which we suppose that is permutable, denote by the elements of the line and the corresponding cofactors, i.e. . This identity will hold for any permutation of ‘s. Therefore, in the case of transpositions we get . If all are equal then obviously, the line is permutable. Suppose now that there are . This means that . Pick another index . Then there exist one of such that , for example. Then . Doing this for all , we see that all cofactors are equal.

Now, if we have two lines which are permutable, there are three cases:

- the two lines contain equal elements, which mean that the determinant is obviously zero.
- if the lines each have equal cofactors, then the classical adjoint has two equal lines, which means that .
- The third case is when we have a line with equal elements and a line with equal cofactors . Then . We can substract from the , and the determinant does not change. , for .