Determinant

Determinant

Definition 1. The determinant of $A \in M_{n \times n}(F)$ is a scalar $$\text{det}(A) = \sum_{j=1}^n (-1)^{i+j}A_{ij}\text{det}(\widetilde{A_{ij}})$$ for some row $i$, where $\widetilde{A_{ij}}$ is the $(n-1) \times (n-1)$ matrix obtained from $A$ by deleting row $i$ and column $j$.
If $n = 1$, then $\det(A) := A_{11}.$

    Determinant, 즉 행렬식은 치환으로 정의되나 여기서는 흔히 라플라스 전개라고 알려진 방법으로 정의하여 잘 알려진 성질들을 소개하는 방식으로 서술하겠다. 

Theorem 1

Theorem 1. Let $u, v, a_i (i = 1, ..., r-1, r+1, ..., n) \in F^n$ and $k \in F$. Then for $1 \leq r \leq n$, we have
$$\det \begin{pmatrix} a_1 \\ \vdots \\ a_{r-1} \\ u+kv \\ a_{r+1} \\ \vdots \\ a_n \end{pmatrix} = \det \begin{pmatrix} a_1 \\ \vdots \\ a_{r-1} \\ u \\ a_{r+1} \\ \vdots \\ a_n \end{pmatrix} + k \det \begin{pmatrix} a_1 \\ \vdots \\ a_{r-1} \\ v \\ a_{r+1} \\ \vdots \\ a_n \end{pmatrix}.$$

Theorem 2

Theorem 2. Let $A, B \in M_{n \times n}(F)$. Then
(a) $\det(AB) = \det(A) \cdot \det(B$).
(b) $\det(A^t) = \det(A$).
(c) $A$ is invertible $\Longleftrightarrow$ $\det(A$) $\neq 0$. Furthermore, if $A$ is invertible, then $\det(A^{-1}$) = $\frac{1}{\det(A)}$.
(d) If rank($A$) $< n$, then $\det(A$) = 0.
(e) If $A$ has a zero row, then $\det(A$) = 0.
(f) If $A$ has two identical rows, then $\det(A$) = 0.
(g) If $A$ is an upper triangular matrix, then $\det(A)$ is the product of its diagonal entries.
(h) If $A$ and $B$ are similar, then $\det(A) = \det(B)$.

$3 \times 3$ 행렬 $$A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$$ 의 determinant는 Levi-Civita symbol과 아인슈타인 합 규약을 사용하여 다음과 같이 표현할 수 있다. $$\det(A) = \varepsilon_{ijk}a_{1j}a_{2j}a_{3k}$$