1) if A, B are square and AB=I_n, then det B=1/det A
2) if A, B, S are square and B=(S^-1)AS, then det B=det A
3) If A is square and (A^T)A=I_n, then det A= +/- 1
4) if A is n x n and A^T= -A, then det A= (-1)^n det A. In particular, if n is odd, then det A=0
SOLUTION This solution is FREE courtesy of BrainMass!
(1) Since , then we have . Thus we must have
(2) Since , then we have
Therefore, we get .
(3) Since , then . Thus we must have
(4) Since , then we have
So we have . Especially when is odd, we have© BrainMass Inc. brainmass.com December 24, 2021, 10:05 pm ad1c9bdddf>