In Michael Artin's Algebra, the discussion on determinant starts from the standard recursive expansion by minors. Artin defines determinant as a function $\delta$ from a square matrix to a real number. Then Artin lists three characteristics for this function in Theorem 1.4.7 (page 20, second edition) as quoted below.
"Theorem 1.4.7 Uniqueness of the Determinant. There is a unique function $\delta$ on the space of $n\times n$ matrices with the properties below, namely the determinant.
- With $I$ denoting the identity matrix, $\delta(I)=1$.
- $\delta$ is linear in the rows of the matrix $A$.
- If two adjacent rows of a matrix $A$ are equal, then $\delta(A)$=0."
In his book, Artin does not explain why $\delta$ should have these properties. I suppose in history people went through a period of trial and error before such abstract concept was proposed and accepted. Can anyone refer me to any source revealing how these properties were thought of, especially, the second and the third property. Thank you! Regards.