Determinant

Theorem 4.4

Det(\(A\))=\(\sum a_{ij}c_{ij}\) where \(c_{ij}\) is the cofactor

Characteristic

• Multilinear
• Alternating (If swap two rows, it will given a minus sign)
• Determinant of Id is 1

Elementary Operations

• Swap: \(*-1\)
• Multiply by \(c\): \(*c\)
• Add a multiple of a row to another row: No impact

Properties

• \(det(AB)=det(A)det(B)\)
• \(A\) is invertible iff \(det(A)\neq 0\)
• \(det(A^T)=det(A)\)

Theorem 4.8

\(det(A)=det(A^T)\)

Proof.

\(rank(A)=rank(A^T)\implies\) if \(det(A)=0, det(A^T)=0\)

\(A=\prod E_i\) (increasing i)

\(A^T=\prod E^T_i\) (decreasing i)

\(\det(A^T)=\prod (det(E^T_i))=\prod (det(E_i))=det(A)\)