Effect of Elementary Row Operations on Determinant

Theorem
Let $\mathbf A = \left[{a}\right]_{n}$ be a square matrix of order $n$.

Let $\det \left({\mathbf A}\right)$ be the determinant of $\mathbf A$.

Take the elementary row operations.


 * $(1): \quad$ Applying $r_i \to ar_i$ has the effect of multiplying $\det \left({\mathbf A}\right)$ by $a$.
 * $(2): \quad$ Applying $r_i \to r_i + ar_j$ has no effect on $\det \left({\mathbf A}\right)$.
 * $(3): \quad$ Applying $r_i \leftrightarrow r_j$ has the effect of multiplying $\det \left({\mathbf A}\right)$ by $-1$.

Proof

 * $(1)$ follows directly from Determinant with Row Multiplied by Constant.
 * $(2)$ follows directly from Multiple of Row Added to Row of Determinant.
 * $(3)$ follows directly from Determinant with Rows Transposed.