Why is the determinant odd when odd numbers are on the diagonal?

[TTob]

I don't understand this :

let A is n x n matrix whose entries are precisely the numbers 1, 2, . . . , n^2.
Put odd numbers into the diagonal of A, only even numbers above the diagonal and arrange the entries under the diagonal arbitrarily. Then det(A) is odd.

What is the explanation ?

 

[HallsofIvy]

What have you tried? In particular, have you tried seeing what happens for n= 2 and 3?

 

[TTob]

for n=2 we have det(A) = -5. so what ?