Proving Matrix Equations: Cramer's Rule, Transpose & Adjoint

[daniel_i_l]

Homework Statement

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
c)adj(gammaA) = gamma^n-1adj(A) if gamma is in R.

Homework Equations

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

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.

Are the answers right? I'm especially hesitant about c).
Thanks!

 

[Dick]

They seem just fine to me.

 

[daniel_i_l]

Thanks a lot!