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