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.

$\blacksquare$