some linear algebra problems i need help with

Let A and B be nxn matrices.
1. Suppose that AB=AC and det A does not equal 0. Show that B=C

2. Show that A is nonsingular if and only if A transpose is nonsingular.

3. Show that det AB = det BA.

4. Show that det AB = 0 if and only if det A=0 or det B=0

5. Show that if AB= -BA and n is odd, then A or B is singular.

6. Show that det A*Atranspose is greater than equal to 0

7. Show that det A*Btranspose = det Atranspose* det B

8. Let A be nxn skew-symmetric matrix. If n is odd, show that det A=0

9. Show that 3x3 vandermonde matrix has a determinant equal to (a-b)(b-c)(c-a) The matrix is
[1 1 1
a b c
a^2 b^2 c^2]
Thank you.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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