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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!!!!

- Thread starter JSG31883
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- #1

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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!!!!

- #2

AKG

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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.

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

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AKG

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(a

(a

(a

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.

- #5

shmoe

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