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..
Equivalence of the system of linear equations Definition 1. Two systems of linear equations are called equivalent if they have the same solution set. Theroem 1 Theorem 1. Let $Ax = b$ be a system of $m$ linear equations in $n$ unknowns, and let $C$ be an invertible $m \times m$ matrix. Then the system $(CA)x = Cb$ is equivalent to $Ax = b$. Proof. Denote $K$ and $K_C$ the solution set to $Ax = b..
Reduced Row Echelon Form Definition 1. A matrix is said to be in reduced row echelon form(RREF) if the following three conditions are satisfied: i) Any row containing a nonzero entry precedes any row in which all the entries are zero. ii) The first nonzero entry in each row is the only nonzero entry in its column. iii) The first nonzero entry in each row is 1 and it occurs in a column to the rig..
