# Orientation preserving and determinants

Can someone help me prove two theorems? I know they both are true, but can't come up with proofs.

1) Prove that a 3x3 matrix A is orientation preserving iff det(A)>0.

2) Prove that for A, B (both 3x3 matrices) that det(AB)=detA*detB. (A, B may or may not be invertible).

THANK YOU!!!!

AKG
Homework Helper
I'm not entirely sure about this one. Let (v w x) be the 3x3 matrix with vectors v, w, and x as columns.

A is orientation preserving

if and only if

det (Av Aw Ax) > 0 iff det (v w x) > 0

if and only if

det (A(v w x)) > 0 iff det (v w x) > 0

if and only if

det(A)det(v w x) > 0 iff det (v w x) > 0 (using number 2. which you need to prove)

if and only if

det(A) > 0

2. I can't think of an easy way to do it, but if you actually expand it out in full, you will be able to show it.

for 2) how can I expand it out? You say if I expand it out I will be able to show it...

AKG
Homework Helper
Take two general matrices, for example, take A to be:

(a11 a12 a13)
(a21 a22 a23)
(a31 a32 a33)

and B to be something similar. Actually compute the product AB and then compute it's determinant, and similarly compute the determinants |A| and |B|, then their product. You'll get some big, long, ugly expressions, but you'll be able to cancel them to show that they're equal.

shmoe