Effect of Elementary Row Operations on Determinant

Theorem
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:

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 $-1$.

Proof
From Elementary Row Operations as Matrix Multiplications, an elementary row operation on $\mathbf A$ is equivalent to matrix multiplication by the elementary matrices corresponding to the elementary row operations.

From Determinant of Elementary Matrix, the determinants of those elementary matrices are as follows:

Exchange Rows
Hence the result.