Homeomorphism calculation help

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The discussion centers on demonstrating that the set of matrices in GL(n;R) with positive determinants is homeomorphic to the set with negative determinants. A proposed method involves replacing the first row of a matrix with its negative, which could serve as an explicit homeomorphism. This transformation maintains the structure of the matrices while changing the sign of the determinant. The conversation emphasizes the importance of understanding the properties of determinants in relation to homeomorphism. The proposed approach highlights a practical technique for establishing the desired homeomorphic relationship.
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How can we show that the set {A in GL(n;R) | det(A)>0} is homeomorphic to the set {A in GL(n;R) | det(A)<0}?"
 
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If you think of A as a matrix, how about just replacing the first row with the negative of itself? Isn't that an explicit homeomorphism?
 

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