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daniel_i_l
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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!
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