Properties of Determinants

The determinant $\det(A)$ has three defining properties:

The Three Properties

1. $\det(I) = 1$
2. Row swap → determinant changes sign
3. Linearity in each row:
   • $\det(\text{row}_i \times c) = c \cdot \det(A)$
   • $\det(\text{row}_i + \text{row}_i') = \det(A) + \det(A')$

Consequences

• Two equal rows → $\det = 0$
• Subtract row multiple → $\det$ unchanged
• Zero row → $\det = 0$
• $A$ triangular → $\det = $ product of diagonal entries
• $\det(AB) = \det(A)\det(B)$
• $\det(A^{-1}) = 1/\det(A)$
• $\det(A^T) = \det(A)$