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## Homework Statement

This problem has two parts:

i) Determine the range of det: O(n) → ℝ.

ii) Are det

^{-1}({1})⊂O(n) and det

^{-1}({1})⊂O(n) groups?

## Homework Equations

AA

^{-1}=I & AA

^{T}=I

## The Attempt at a Solution

i) det(AA

^{T})=det(I)

det(AA

^{T})=1

det(A) det(A

^{T})=1

det(A) det(A)=1

(det(A))

^{2}=1

det(A)= +-1

So the range is +-1.

**Did I make any mistakes?**

ii) The function det: O(n) → ℝ is not injective, so therefore the inverse function does not exist (right?). Which would result in det

^{-1}({1})⊂O(n) and det

^{-1}({1})⊂O(n) not being groups (right?).

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