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Let E bet the set of matrices A in M(2n,R) such that

\( a_{ii}=0 \) for any \( 1\leq i\leq 2n \)

\( a_{ij} \) is -1 or 1 for any \( 1\leq i\neq j\leq 2n \)

1/ For A in E prove det(A) is an odd integer.

2/ Find the maximum of \( |\det(A)| \) for any matrix \( A \in E \) and find the minimum of \( |\det(A)| \) for any \( A \in E \).

1) Pentru paritatea determinantului, observam ca nu conteaza daca schimbam un semn. (daca ne uitam, de exemplu in formula determinantului). Atunci putem considera matricea cu toate elementele 1 in afara de diagonala primcipala, unde sunt 0.

Acum adunam toate coloanele la prima coloana, si obtinem \( 2n-1 \) pe prima coloana peste tot. Dam factor pe \( 2n-1 \), si obtinem ca determinantul matricii noastre are aceeasi paritate cu matricea initiala, la care punem un 1 in coltul de sus, in stanga. Scadem prima coloana din celelalte si obtinem (in afara de prima coloana care ramane 1) -1 pe diagonala principala si 0 in rest, deci determinantul este -1, care este impar. Deci si matricea initiala are determinantul impar.

(we can consider all elements 1 and on the diagonal 0; this doesn't affect the parity of the determinant. Then we add all colunms to the first and factor \( 2n-1 \). Now we have a matrix with the first column 1. We substract this column from the others and we obtain -1 on the diagonal and 0 in rest, so the determinant is -1, odd, so the initial determinant is odd)


Imi pare rau pentru solutia povestita, dar nu am vreme sa scriu matrici asa mari acuma