(adsbygoogle = window.adsbygoogle || []).push({}); 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

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

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.

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

Thanks!

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# Homework Help: Proofs with matrices

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