# Homework Help: Proofs with matrices

1. Apr 12, 2007

### daniel_i_l

1. The problem statement, all variables and given/known data
Prove or disprove the following:
(A is a nxn square matrix)
a) The vector b is in R^n and all its elements are even integers. If all the elements of the A are integers and det(A) = 2, then the equation Ax = b has a solution with only integer elements
b) If n is odd and transpose(A) = -A then the equation Ax=0 has only one solution

2. Relevant equations
a) Cramer's rule.
b) det(transpose(A)) = det(A)
det(-A) = (-1)^ndet(A)
c) definition of adjoint (transpose of cofactor matrix)

3. The attempt at a solution

a)True: according to Cramer's rule each element of the solution is the determinant of A with one coloum replaced by b divided by det(A). Since the determinant can be taken with coloum b the first determinant will be the sum of even numbers (because all of b's elements are even and A's are integers) and that divided by |A|=2 will be an integer.

b)False: |transpose(A)| = |A|. |-A| = (-1)^nA = -|A| cause n is odd. So 2|A|=0 => |A|=0 and A isn't reversable so it must have a non trivial solution to Ax=0.

c)True: because each element of adj(gammaA) is +/-1 times a determinant of gamma * (a n-1xn-1 matrix) which equals gamma^n-1 times the determinant of the n-1xn-1 matrix.

Thanks!

Last edited: Apr 12, 2007
2. Apr 12, 2007

### Dick

They seem just fine to me.

3. Apr 12, 2007

### daniel_i_l

Thanks a lot!