------Plan
Finish 4.5
Compute some determinants if time
4.5
Alternating implies switching rows give a minus sign.
Corollary: if any two rows are the same, then the determinant is zero.
n-linear implies scaling a row scales the determinant.
If a matrix is singular then the determinant is zero.
Adding a multiple of 1 row to another does not change rank.
det(AB)=det(A)det(B) break into elementary
det(A)= prod det(Ei) = prod det(Etranspose) = det(Atranpose) break into elementary
Any 2 alternating n-linear functions that are 1 on the identity are the same. This proves uniqueness. It remains to prove existence.
4.2
The determinant is defined recursively. Let tilde Aij be the matrix formed by deleting the i-th row and j-th column.
determinant is cofactor expansion along the first row. The cofactor is the signed sub determinant term.
det(A) = sum A1j c1j
determinant is n-linear.