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.