Effect_of_Elementary_Row_Operations_on_Determinant

Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$. Let $\map \det {\mathbf A}$ denote the determinant of $\mathbf A$. Take the elementary row operations {{axiom | n = \text {ERO} 1 | t = For some $\lambda$, multiply row $i$ by $\lambda$ | m = r_i \to \lambda r_i }} {{axiom | n = \text {ERO} 2 | t = For some $\lambda$, add $\lambda$ times row $j$ to row $i$ | m = r_i \to r_i + \lambda r_j }} {{axiom | n = \text {ERO} 3 | t = Exchange rows $i$ and $j$ | m = r_i \leftrightarrow r_j }} Applying $\text {ERO} 1$ has the effect of multiplying $\map \det {\mathbf A}$ by $\lambda$. Applying $\text {ERO} 2$ has no effect on $\map \det {\mathbf A}$. Applying $\text {ERO} 3$ has the effect of multiplying $\map \det {\mathbf A}$ by $-1$.