16.7 Properties of Determinants

Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: 16.5 Permutation Definition of Determinant

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Properties of Determinants

Let $A, B \in R^{n \times n}$ and $c \in R$ . Then, the determinant function $det (A)$ given by

det (A) = \sum_{π \in S_{n}} sgn (π) \cdot a_{1 π (1)} \cdot a_{2 π (2)} \dots a_{n π (n)}

satisfies all of the following:

If $A$ is 8.9 Upper-Triangular Matrix or 8.7 Lower-Triangular Matrix, then $det (A) = a_{11} \cdot a_{22} \dots a_{n n}$
$det (A) = det (A^{T})$
$det (A \cdot B) = det (A) \cdot det (B)$
If $S$ is invertible (nonsingular), then $det (S \cdot A \cdot S^{- 1}) = det (A)$
If $i \neq k$ , then $det (P_{i k} \cdot A) = - det (A)$
If $i \neq k$ , then $det (S_{i k} (c) \cdot A) = det (A)$
For $1 \leq i \leq n, det (D_{i} (c) \cdot A) = c \cdot det (A)$
$det (c A) = c^{n} det (A)$
$A$ is invertible iff $det (A) \neq 0$