Orientation preserving and determinants

[JSG31883]

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]

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.

 

[JSG31883]

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

 

[AKG]

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]

Expanding out 2) will be disgusting (but would certainly work). Another way is to first prove it for elementary matrices, then show that any invertible matrix is the product of elementary matrices and you've pretty much handled the invertible case. A or B non-invertible is easier, assuming you know non-invertible<=>determinant is zero (or can prove this).