Prove that $\det(A)\neq 0$.

Question

Let $A$ be a $n \times n$ matrix, $n$ even, with even diagonal elements and all other elements odd integers. Prove that $\det(A)\neq 0$. Can anyone give me a hint? Thank you.

Use Laplace's formula to prove with induction, that $\det A$ is odd. — Tomas, Aug 15 '13 at 08:10

score 14 · Answer 1 · answered Aug 15 '13 at 08:00

14

Hint: Compute the determinant of $A$ reduced modulo $2$.

answered Aug 15 '13 at 08:00

Seirios

33,157

score 1 · Answer 2 · answered Aug 15 '13 at 08:02

1

The determinant is an odd integer then.

answered Aug 15 '13 at 08:02

Marc Palm

4,641

Prove that $\det(A)\neq 0$.

2 Answers2