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 $\det(B) = \det(A)$.