Cramer's Rule Theorem 1. (Cramer's Rule) Let $Ax = b$ be a system of $n$ linear equations in $n$ unknowns, where $x = (x_1, ..., x_n)^t$. If $\det(A) \neq 0$, then this system has a unique solution, and $$x_k = \frac{\det(M_k)}{\det(A)}, \forall k \in \{1, ..., n\},$$ where $M_k \in M_{n \times n}(F)$ obtained from $A$ by replacing column $k$ of $A$ by $b$. Proof. Let $y \in F^n$, and let denote..
What Happens to $\det(A)$ if we perform an elementary row operation on $A$ Theorem 1. Let $A \in M_{n \times n}(F)$ and $B = R(A)$, where $R$ is an elementary row operation. Then the followings hold: (a) If $R = R_{i \leftrightarrow j}$, then $\det(B) = -\det(A)$. (b) If $R = R_{ci}$, then $\det(B) = c \cdot \det(A)$. (c) If $R = R_{i + cj}$, then $\det(B) = \det(A)$.
이 포스트에서 $V$는 유한차원 $F$-벡터공간으로 취급한다. Determinant of a Linear Operator Definition 1. Let $T \in \mathcal{L}(V)$. We define the determinant of $T$, denoted $\det(T)$, to be $\det(T) = \det([T]_{\beta})$, where $\beta$ is an ordered basis for $V$. 선형 연산자 $T$의 행렬 표현의 행렬식으로 $T$의 행렬식을 정의할 수 있다. 이러한 정의는 $V$의 기저의 선택에 의존하지 않는다. Let $\beta, \gamma$ be ordered bases for $V$. By Theorem 2, we have $[T]_{\gamm..
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..
Least Common Multiple Definition 1. Let $a, b \in \mathbb{Z}$, with $a \neq 0, b \neq 0$. The least common multiple of $a$ and $b$, denoted by lcm($a, b$), is $m \in \mathbb{N}$ satisfying the following: (a) $a \,|\, m \wedge b \,|\, m$. (b) $a \,|\, c \wedge b \,|\, c (c > 0) \Longrightarrow m \leq c$. 최대공약수와 마찬가지의 방법으로 lcm, 즉 최소공배수를 정의할 수 있다. 공배수이면서 ((a)) 공배수 중 가장 작은 수를 ((b)) 최소공배수라고 한다. Theor..
