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In book of Emil Artin "Geometric Algebra" i found such definition of determinant Det: function on matrix rows $A_{1}, ..., A_{n}$, that satisfies:

  • $Det(A_{1}, ..., b*Ai,..., A_{n})=b*Det(A_{1}, ..., Ai,..., A_{n})$
  • $Det(A_{1}, ..., Ai+b*Aj,..., A_{n})=b*Det(A_{1}, ..., Ai,..., A_{n})$ - adding scaled row to another row not changing matrix.
  • For Identity matrix $Det(I)=1$

In the book its shown directly, that such function exists. Its also prooved there, that in comutative case (field), such function is multilinear. So, its easy to see, that in commutative case such defenition of determinant is equivalent to standart defenition of multilinear alternating map - if rows dublicated, we can substract one from another, and push zero out - so Artin determinant is alterating. It's shown here, that "standart" det applied to transvection gives 1. I know that standart det is unique function for field - here and here.

I wonder if Artin Det is unique, and how to proove it directly - non-direct proof is given above for fields, but there is still question for general case of division rings.

nagvalhm
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  • Actually, i realized proof, just after publication. It follows directly from that any matrix presented as $E_{1}....E_{k}D_{x}$ - multiple of transvections and diagonal matrix D(x) - identity matrix with x in last column - all in the book. I don't delete question, may be it would be helpful for someone. – nagvalhm Aug 24 '23 at 14:46
  • If I understand it correctly, this has already been asked and answered here and here. – Jacob Maibach Aug 24 '23 at 16:08
  • @Jacob Maibach, no, there are different det-definition and different Artin mentioned, actualy. I gave proof, that these definitions are equivalent, but only for GL over fields. – nagvalhm Aug 24 '23 at 16:17
  • @Jacob Maibach, but what you gave, was mentioned in my question without links, so thanks. – nagvalhm Aug 24 '23 at 16:22

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