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.