Module: Determinants

Learning Outcomes

Determinants

• Look for functions from square matrices to numbers satisfying three axioms (linear function of any row, if all others are held fixed; changes sign if rows exchanged; identity matrix assigned 1)
• Only one function satisfies all—that function is the determinant
• If rows are linearly dependent, determinant is 0 (from axioms)
• Determinant of lower/upper triangular or diagonal matrices: product of diagonal entries (from axioms)
• determinant = product of pivots
• Square \(A\) invertible iff determinant(\(A\)) \(=0\) (follows from previous)
• determinant\((AB)\) = determinant\((A)\) determinant\((B)\) (from axioms)
• determinant\((A)\) = determinant\((A^T)\) (from elimination written in matrix form)
• Recursive formula for determinants (from axioms)
• Cramer’s rule

Readings

Experiential Learning