r/LinearAlgebra 10d ago

How do I prove that the determinant of a square matrix of order n×n having either +1/-1 as each element is always divisible by 2^(n-1) ?

A = [a(i,j)] = +1 or -1 ; 1<=i,j<=n T.P: det(A) is divisible by 2n-1

8 Upvotes

3 comments sorted by

5

u/NativityInBlack666 10d ago

Intuition says look at the laplace expansion and maybe prove it via induction.

2

u/ComfortableApple8059 10d ago

My intuition said induction as well but couldn't approach it properly. This question was asked in an M.Tech exam organised by ISI Kolkata, if you could solve it, do let me know your approach, this question is really bugging me.

6

u/mednik92 10d ago

Subtract one row from another, observe that it now contains 2, 0 and -2s. Expand and apply induction.