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Enjoicube

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## Homework Statement

Prove that the determinant of a unitary matrix is +/-1

## Homework Equations

<Av,Aw>=<v,w>

Det(AB)=Det(A)*Det(B)

## The Attempt at a Solution

Alright, I am aware that <Av,Aw>=<v,w> => A(tA)=I and (tA)A=I so Det(A(tA))=Det(I)=1 thus

Det(A)*Det(tA)=1. However, this is where I am stuck, I am aware that in some cases Det(tA)=Det(A), however this is not obvious in this case, as I cannot take tA to be the elementary defintion of switching rows and columns. How can I prove that Det(tA)=Det(A) ONLY from the fact that <Aw,v>=<w,(tA)v>. This inner product can be absolutely any symmetric bilinear form, so do not assume I mean the dot product.

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