## 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.